RF stripline circulator devices and methods

ABSTRACT

Microwave circulators are an essentially component in many microwave systems and whilst the waveguide technologies they are implemented in have evolved their design today still employs procedures that are typically approximate and have no regular approach, which in many instances is through dependence on empirical equations or considered a trade secret that gives an edge to commercial suppliers of microwave and RF circulators. The result is expensive isolators where high performance is required as they are merely selected out or require manual tuning. Further, for broadband systems, designer&#39;s resort to dividing into sub-bands deploying multiple narrower band circulators. Accordingly, the inventors present a design methodology based on an accurate closed form solution allowing the selection of suitable ferrite specifications for the required operating bandwidth as well as calculating the ferrite disc impedance allowing the necessary matching network to be designed and the circulator design completed.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent application claims the benefit of priority fromInternational Patent Application PCT/CA2018/000,027 entitled “RFStripline Circulator Devices and Methods” filed Feb. 16, 2018; whichitself claims the benefit of priority from U.S. Provisional PatentApplication 62/460,183 entitled “RF Stripline Circulator Devices andMethods” filed Feb. 17, 2017; the entire contents of each beingincorporated herein by reference.

FIELD OF THE INVENTION

This invention relates to both RF stripline circulators and to ridge gapcirculators and a systematic design procedure—a methodology for both,stripline and ridge gap circulators with an intentionally designed airgap around the ferrite disc.

BACKGROUND OF THE INVENTION

Microwave circulators were proposed 60 years ago, to be deployed indifferent communication systems and radar applications and have gonethrough substantial development subsequently. Initially, circulatorswere designed according to Faraday rotation and were developed for highpower handling devices such as resonance circulators and differentialphase shift circulators. One of the most important configurations is thejunction circulator, also known as a turnstile circulator orY-circulator. The configuration of this circulator is formed using aY-shaped structure with three identical guiding structures in the middleof which a ferrite disc is located. This middle section provides thenonreciprocal characteristics of the Y-junction. The three identicalarms can be rectangular waveguides, striplines, microstrip lines, or anyguiding structure. The objective over this period of time forRF-microwave circuit designers being to achieve improved electricalspecifications within a smaller footprint and/or lower cost. Recently,modern guiding structures have been established for circulatorimplementations such as those employed within Substrate IntegratedWaveguide (SIW) and the coupled line circulators. Modern fabricationtechniques are now also utilized to produce semiconductor based andMIMIC based circulators, as well.

Amongst these, the new guiding structures that require components basedon its technology is the ridge gap waveguide (RGW), which was introducedrecently in 2009. The concept of this configuration builds on theconcepts of soft and hard surfaces wherein the design methodology is tohave full confinement of the microwave (RF) signal between two parallelplate like structures. The signal leakage is eliminated by the existenceof a two-dimensional (2D) soft surface surrounding the signal path,which forms an Artificial Magnetic Conductor (AMC). The AMC with thePerfect Electrical Conductor (PEC), the upper ground of the RGW,prevents the signal from propagating (leaking) outside the ridge. Thebasic concepts of the RGW waveguide have been addressed and tested inmany publications within the prior art. One of the advantages of the RGWstructure is its broad operating bandwidth that can exceed 3:1 in someprior art embodiments.

However, within this prior art, which has yielded many models for thejunction analysis to describe the characteristics of the junctionsmathematically and used multiple numerical techniques in achievingcirculator solutions, the procedures are typically approximate and haveno regular approach, which in many instances is through dependence onempirical equations or considered a trade secret that gives an edge tocommercial suppliers of microwave and RF circulators. Thus, the designsolutions available within the prior art are inaccurate and lead toproduction process that pass through an iterative process between thetesting laboratory and the machine shop (and other processing stages) inaddition to which wideband circulators have to post-fabrication tuned.As a result, system engineers tend to divide the operating bandwidthinto sub-bands deploying multiple narrower band circulators within thesystem in an attempt to overcome these issues. Further, the impedancematching section for the circulator junction either exploits expensive“special composite materials” and/or requires post-fabricationprocessing.

Accordingly, it would be beneficial to provide component and circuitdesigns with a systematic approach to designing circulators with closedform expressions. It would be further beneficial for such closed formexpressions to be applicable to any frequency band with proper scaling.Such a methodology allowing for reductions in the time required fordesign, fabrication, and testing, and the cost of each circulator whilstenabling the availability of ultra-wideband circulators with reasonablecosts and production logistics.

Within this specification, the inventors present such a designmethodology based on an accurate closed form solution allowing theselection of suitable ferrite specifications for the required operatingbandwidth as well as calculating the ferrite disc impedance allowing thenecessary matching network to be designed. Further, the inventors haveestablished an alternative circulator design employing:

-   -   A dielectric filling within the RGW to match the waveguide        striplines feeding to the center disc;    -   A perforated substrate to achieve the required effective        permittivity for impedance matching;    -   Employing a standard substrate to achieve the lowest possible        cost; and    -   Employing an air gap around the ferrite disc to minimize the        fringing fields and bound the effective diameter of the disc        close to its physical diameter.

Based on these design algorithms and fabrication considerationsultra-wideband circulator production may be achieved with reduced timeand cost with enhanced performance characteristics. Exemplaryimplementations with respect to two designs centered at 15 GHz and 30GHz respectively, for 5G mobile applications are presented.

Other aspects and features of the present invention will become apparentto those ordinarily skilled in the art upon review of the followingdescription of specific embodiments of the invention in conjunction withthe accompanying figures.

SUMMARY OF THE INVENTION

It is an object of the present invention to mitigate limitations withinthe prior art relating to RF stripline circulators and to ridge gapcirculators and a systematic design procedure—a methodology for both,stripline and ridge gap circulators with an intentionally designed airgap around the ferrite disc.

In accordance with an embodiment of the invention, there is provided amethod of designing a microwave circulator comprising:

-   1) Solving a predetermined set of closed form equations at a    predetermined frequency relating to the electrical and magnetic    fields with respect to an electrically non-conductive and    ferromagnetic element comprising a first predetermined portion of    the microwave circulator; and-   2) Designing a matching transformer to cover a predetermined    bandwidth of operation depending on the simulation data established    in step (1), the simulation data comprising a set of physical    properties of the electrically non-conductive and ferromagnetic    element, a set of physical properties of a plurality of microwave    ports forming a second predetermined portion of the microwave    circulator and a set of electrical properties of the microwave    ports.

In accordance with an embodiment of the invention, there is provided amicrowave circulator comprising:

-   a pair of electrically non-conductive and ferromagnetic elements    with specific magnetic saturation value (M_(s)) each having a    predetermined thickness and a predetermined diameter;-   an electrical conductor plane comprising a plurality of microwave    tracks and a central circular pad to which each microwave track is    coupled at a predetermined location, each microwave track comprising    a first portion adjacent the central pad and a second portion    extending from the first portion to a distal point;-   a lower electrical ground plane;-   an upper electrical ground plane;-   a first dielectric disposed between the electrical conductor plane    and the lower electrical ground plane and having a thickness    determined in dependence upon the predetermined thickness of the    electrically non-conductive and ferromagnetic elements and an    opening determined in dependence upon the predetermined diameter of    the electrically non-conductive and ferromagnetic elements;-   a second dielectric disposed between the electrical conductor plane    and the upper electrical ground plane and having a thickness    determined in dependence upon the predetermined thickness of the    electrically non-conductive and ferromagnetic elements and an    opening determined in dependence upon the predetermined diameter of    the electrically non-conductive and ferromagnetic elements; wherein-   the openings within the first dielectric and second dielectric have    a diameter establishing a predetermined air gap between the external    periphery of an electrically non-conductive and ferromagnetic    element and their respective dielectric when the electrically    non-conductive and ferromagnetic element is centrally disposed of    with the opening;-   the first portion of each microwave track is air filled microwave    track; and-   the second portion of each microwave track is a dielectric filled    microwave track.

In accordance with an embodiment of the invention, there is provided amicrowave circulator comprising:

-   a set of three parallel electrical planes wherein the middle    electrical plane comprises a plurality of microwave tracks and a    central region coupled to the plurality of microwave tracks and each    outer electrical plane is a ground plane; wherein-   a central portion of the set of three parallel electrical layers    comprises an inner region with electrically non-conductive and    ferromagnetic elements of predetermined lateral dimensions disposed    between each outer electrical plane and the middle electrical plane    and an outer region filled with a first dielectric material of low    dielectric constant such that those portions of each microwave track    in this outer region form microwave feeds coupled to the central    region of the middle electrical plane at predetermined locations;-   an outer portion of the set of three parallel electrical layers is    filled with a second dielectric material such that those portions of    each microwave track in this outer portion form microwave matching    networks between the part of each microwave track in the outer    region of the central portion and an external microwave circuit to    be coupled to the distal ends of each microwave track from the    central portion.

In accordance with an embodiment of the invention, there is provided amethod of designing a microwave circulator comprising:

-   1) solving a predetermined set of closed form equations at a    predetermined frequency relating to the electrical and magnetic    fields with respect to an electrically non-conductive and    ferromagnetic element comprising a first predetermined portion of    the microwave circulator with low dielectric constant material based    microwave waveguides coupling to the electrically non-conductive and    ferromagnetic element; and-   2) designing a matching transformer to cover a predetermined    bandwidth of operation in dependence upon simulation data    established in step (1) using high dielectric constant substrate    based microwave waveguides forming a matching network between the    waveguides coupling to the electrically non-conductive and    ferromagnetic element and an external microwave circuit coupled to    the microwave circulator, the simulation data comprising a set of    physical properties of the electrically non-conductive and    ferromagnetic element, a set of physical properties of a plurality    of microwave ports forming a second predetermined portion of the    microwave circulator and a set of electrical properties of the    microwave ports.

Other aspects and features of the present invention will become apparentto those ordinarily skilled in the art upon review of the followingdescription of specific embodiments of the invention in conjunction withthe accompanying figures.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will now be described, by way ofexample only, with reference to the attached Figures, wherein:

FIG. 1 depicts the Gytropy versus the normalized Larmor frequency for aferrite disc saturated in the {circumflex over (z)} direction;

FIG. 2 depicts a schematic of a junction circulator according to anembodiment of the invention;

FIGS. 3A and 3B depict the simulated model for a circulator according toan embodiment of the invention exploiting straight arms in top view andthree-dimensional (3D) model respectively;

FIGS. 3C to 3F depict the simulated model for a circulator according toan embodiment of the invention exploiting curved arms in top views andthree-dimensional (3D) models;

FIGS. 3G and 3H depict the simulated model for a circulator according toan embodiment of the invention exploiting two curved arms inthree-dimensional (3D) model respectively focusing on the PerfectElectrical Conductor (PEC) and Perfect Magnetic Conductor (PMC) boundaryconditions respectively;

FIG. 4 depicts a comparison between theoretical and simulated responsesfor the exemplary K-band circulator according to an embodiment of theinvention with α=0;

FIGS. 5A and 5B depict the exemplary K-band circulator according to anembodiment of the invention with perforations in plan and 3D viewsrespectively;

FIG. 6 depicts the scattering parameters for an exemplary K-bandcirculator according to an embodiment of the invention with perforatedmatching transformer and α=0;

FIG. 7 depicts a comparison between theoretical and simulated responsesfor the exemplary K-band circulator according to an embodiment of theinvention with α=0.2;

FIG. 8 depicts the scattering parameters for the exemplary K-bandcirculator according to an embodiment of the invention with perforatedmatching transformer and α=0.2;

FIG. 9 depicts a comparison between theoretical and simulated responsesfor the exemplary X-band circulator according to an embodiment of theinvention with α=0;

FIG. 10 depicts a comparison between theoretical and simulated responsesfor the exemplary X-band circulator according to an embodiment of theinvention with α=0.2;

FIGS. 11A and 11B depict the exemplary K-band circulator according to anembodiment of the invention with perforations in plan and 3D viewsrespectively;

FIG. 12 depicts scattering parameters for an exemplary X-band circulatoraccording to an embodiment of the invention with perforated matchingtransformer and α=0;

FIG. 13 depicts the scattering parameters for the exemplary X-bandcirculator according to an embodiment of the invention with perforatedmatching transformer and α=0.2;

FIG. 14A depicts the axial electric field within the exemplary X-bandcirculator according to an embodiment of the invention at f=23.3 GHz;

FIG. 14B depicts the axial electric field within the exemplary K-bandcirculator according to an embodiment of the invention at f=10.82 GHz;

FIGS. 15A and 15B depict the comparison between the analytical and thesimulated response of the RGW circulators with ideal PMC around theridge for the 15 GHz and 30 GHz designs respectively;

FIGS. 16A and 16B depict a front view and 3D view of the “bed of nails”unit cell for the RGW structure;

FIGS. 16C and 16D depict the dispersion relationships for the “bed ofnails” unit cells for the RGW structure at 15 GHz and 30 GHzrespectively;

FIG. 17 depicts a 3D view of the realized RGW circulator exploiting the“bed of nails”

FIGS. 18A and 18B depict scattering parameters for the 15 GHz and 30 GHzRGW circulators respectively.

DETAILED DESCRIPTION

The present invention is directed to RF stripline circulators and toridge gap circulators and a systematic design procedure—a methodologyfor both, stripline and ridge gap circulators with an intentionallydesigned air gap around the ferrite disc.

The ensuing description provides representative embodiment(s) only andis not intended to limit the scope, applicability or configuration ofthe disclosure. Rather, the ensuing description of the embodiment(s)will provide those skilled in the art with an enabling description forimplementing an embodiment or embodiments of the invention. It is beingunderstood that various changes can be made in the function, andarrangement of elements without departing from the spirit and scope asset forth in the appended claims. Accordingly, an embodiment is anexample or implementation of the inventions and not the soleimplementation. Various appearances of “one embodiment”, “an embodiment”or “some embodiments” do not necessarily all refer to the sameembodiments. Although various features of the invention may be describedin the context of a single embodiment, the features may also be providedseparately or in any suitable combination. Conversely, although theinvention may be described herein in the context of separate embodimentsfor clarity, the invention can also be implemented in a singleembodiment or any combination of embodiments.

Reference in the specification to “one embodiment”, “an embodiment”,“some embodiments” or “other embodiments” means that a particularfeature, structure, or characteristics described in connection with theembodiments is included in at least one embodiment, but not necessarilyall embodiments, of the inventions. The phraseology and terminologyemployed herein are not to be construed as limiting, but is fordescriptive purpose only. It is to be understood that where the claimsor specification refer to “a” or “an” element, such reference is not tobe construed as there being only one of that element. It is to beunderstood that where the specification states that a component feature,structure, or characteristics “may”, “might”, “can” or “could” beincluded, that particular component, feature, structure, orcharacteristics are not required to be included.

Reference to terms such as “left”, “right”, “top”, “bottom”, “front” and“back” are intended for use in respect to the orientation of theparticular feature, structure, or element within the figures depictingembodiments of the invention. It would be evident that such directionalterminology with respect to the actual use of a device has no specificmeaning as the device can be employed in a multiplicity of orientationsby the user or users. Reference to terms “including”, “comprising”,“consisting” and grammatical variants thereof do not preclude theaddition of one or more components, features, steps, integers or groupsthereof, and that the terms are not to be construed as specifyingcomponents, features, steps or integers. Likewise, the phrase“consisting essentially of”, and grammatical variants thereof, when usedherein is not to be construed as excluding additional components, steps,features, integers or groups thereof, but rather that the additionalfeatures, integers, steps, components or groups thereof do notmaterially alter the basic and novel characteristics of the claimedcomposition, device or method. If the specification or claims refer to“an additional” element, that does not preclude there being more thanone of the additional elements.

1. Background—Normalized Magnetic Factors and Gytropy

Expressions for both normalized magnetic factors are listed below inTable 1 and are addressed in more detail below.

TABLE 1 Permeability Tensor Important Expressions The Physical QuantitySymbol Expression Gytropy k/μ$\frac{\omega\mspace{14mu}\omega_{0}}{\omega_{0}^{2} + \omega^{2} + {\omega_{0}\omega_{m}}}$Diagonal elements μ μ₀(1 + Ψ_(xx)) = μ₀(1 = Ψ_(yy)) Off diag. elements k−jμ₀Ψ_(xy) = jμ₀Ψ_(yx) Diagonal susceptibility Ψ_(xx)$\frac{\omega_{0}\omega_{m}}{\omega_{0}^{2} - \omega^{2}}$ Larmorangular frequency ω₀ μ₀γ“H₀(A/m)” 2π * 2.8 * 10⁶ “H₀(O_(e))” Magneticangular frequency ω_(m) μ₀γ“M₀(A/m)” 2π * 2.8 * 10⁶ “(4πM_(s))(G)”Normalized Larmor frequency σ₀$\frac{\omega_{0}}{\omega} = \frac{2.8*10^{6\mspace{14mu}}{``{H_{0}\left( O_{e} \right)}"}}{f}$Normalized magnetic frequency p_(m)$\frac{\omega_{m}}{\omega} = \frac{2.8*10^{6\mspace{14mu}}{``{\left( {4\pi\; M_{s}} \right)(G)}"}}{f}$

The normalized Larmor frequency is denoted by σ₀, while the normalizedmagnetic frequency is denoted by p_(m). Both are normalized to theoperating frequency. The modes of any circulator operation can be showneasily based on these factors. The normalized Larmor frequency σ₀ can bealso written in terms of the resonance magnetic field and the actualbias as defined by Equation (1).

$\begin{matrix}{\sigma_{0} = {\frac{\omega_{0}}{\omega} = {\frac{2.8*{10^{6}}^{''}{H_{0}\left( O_{e} \right)}^{''}}{f} = \frac{H_{0}}{H_{r}}}}} & (1)\end{matrix}$

This makes σ₀=1 is the boundary between the above resonance mode and thebelow resonance mode. The negative values of σ₀ corresponds to negativevalue of H₀. This occurs only when the applied external magnetic fieldis not sufficient for saturation. In this case the losses due tounsaturated ferrites will be dominant. The Gytropy (g_(k)=k/μ) can berelated to the normalized magnetic factors through Equation (2).

$\begin{matrix}{g_{k} = \frac{p_{m}}{\sigma_{0}^{2} + {\sigma_{0}p_{m}} - 1}} & (2)\end{matrix}$

This equation is plotted for different p_(m) values in FIG. 1 and allmodes of operations are indicated in FIG. 1. The region before ferritesaturation is, also, indicated in FIG. 1, where the losses increasedramatically. This region is not an operating mode for any circulator.The resonance mode exits around the value of σ₀=1, where it is indicatedroughly in this FIG. 1. To indicate this region exactly, the line widthof the material should be given and it can be indicated by Equation (3).

$\begin{matrix}{{1 - \frac{\Delta\; H}{H_{r}}} < \sigma_{0} < {1 + \frac{\Delta\; H}{H_{r}}}} & (3)\end{matrix}$

This curve in FIG. 1 is crucial in the design procedure as thecirculator specifications are related to the Gytropy |g_(k)|. Based onthe required value of |g_(k)|, this curve is utilized to select thesuitable σ₀ and p_(m). This leads to the choice of the utilized ferritematerial and the applied bias. Another prospective in the designprocedure is to assume a relation between the saturation magnetizationof the material and the applied magnetic field as given by Equation (4)which will be the same ratio between the normalized Larmor frequency andthe normalized magnetic frequency in Equation (5), where, α is calledthe magnetic biasing ratio. Dealing with α as a design parameter,reduces the number of unknowns by one.

$\begin{matrix}{{H_{0}\left( O_{e} \right)} = {4\pi\;{M_{s}(G)}*\alpha}} & (4) \\{\sigma_{0} = {\alpha\; p_{m}}} & (5) \\{g_{k} = \frac{p_{m}}{{\left( {\alpha^{2} + \alpha} \right)p_{m}^{2}} - 1}} & (6) \\{p_{m} = {\frac{1}{2\left( {\alpha^{2} + \alpha} \right)g_{k}}\left\lbrack {1 \pm \sqrt{1 \pm {4\;{g_{k}^{2}\left( {\alpha^{2} + \alpha} \right)}}}} \right\rbrack}} & (7)\end{matrix}$

Rewriting the relation between the Gytropy and the normalized magneticfactors yields Equation (6) which can be reformed as Equation (7).Therefore, by selecting the magnetic biasing ratio, the material can bedetermined to achieve the required Gytropy. It is important to mentionthat the majority of the circulator designers start their analysis withthe assumption of having α=0. This is based on the assumption that thematerial is just saturated. This assumption is practically invalid. Themagnets deployed to provide the DC magnetic biasing are usuallypermanent magnets. There is no practical methodology to increase themagnetic field in a continuous way. Initially, the magnetic bias startswith a top and a bottom magnet. It is possible to add one magnetic discor two discs. Sometimes, it is possible to add a smaller magnet disc,however, the design consideration of having exact saturation iscritical. Losing the required biasing point in this case leads to go inthe low field losses region.

2. Junction Circulator Mathematical Formulations

As is normal in electromagnetic problems the starting point areMaxwell's equations, which can be written in the source free case asEquations (8) and (9) where, in the case of the ferrite material thepermeability is represented by a tensor given by Equation (10), whichassumes the ferrite disc is saturated in the z-direction, can beexpressed as given by Pozar in “Microwave Engineering” (John Wiley &Sons, 3rd Edition, 2005) in Equation (11).

$\begin{matrix}{{\nabla{\times \overset{\_}{E}}} = {{- j}\;\omega\;\overset{\_}{B}}} & (8) \\{{\nabla{\times \overset{\_}{H}}} = {{- j}\;\omega\;\overset{\_}{D}}} & (9) \\{\overset{\_}{B} = {\lbrack\mu\rbrack\overset{\_}{H}}} & (10) \\{\lbrack\mu\rbrack = \begin{bmatrix}\mu & {j\;\kappa} & 0 \\{{- j}\;\kappa} & \mu & 0 \\0 & 0 & \mu_{0}\end{bmatrix}} & (11)\end{matrix}$

This representation is valid in both rectangular and cylindricalcoordinate representations. The expressions used to calculate μ and kare summarized in Table 1. Hence, these equations can be rewritten asEquations (12) and (13).∇×Ē=−jω[μ] H   (12)∇× H=−jωεĒ  (12)

Maxwell equations are solved in the cylindrical coordinates, taking intoconsideration the permeability tensor of a saturated ferrite in thez-direction. The axial variation is assumed to be zero, i.e. ∂/∂z=0.Solving Equation (12) yields Equations (14) and (15) and solving theprevious two equations together, the transverse magnetic fieldintensities can be related to the axial electric field through thefollowing Equations (16) and (17), respectively, where the effectivewave number, the effective intrinsic admittance and the effectivepermeability are expressed by Equations (18) to (20), respectively.

$\begin{matrix}{{\frac{1}{\rho}\frac{\delta\; E_{z}}{\delta\;\phi}} = {{- j}\;{\omega\left( {{\mu\; H_{\rho}} + {j\;\kappa\; H_{\phi}}} \right)}}} & (14) \\{{- \frac{\delta\; E_{z}}{\delta\;\rho}} = {{- j}\;{\omega\left( {{{- j}\;\kappa\; H_{\rho}} + {\mu\; H_{\phi}}} \right)}}} & (15) \\{H_{\rho} = {\frac{j\; Y_{0\;{eff}}}{k_{eff}\mu}\left( {{\mu\frac{\delta\; E_{z}}{\delta\;\rho}} + {\frac{\mu}{\rho}\frac{\delta\; E_{z}}{\delta\;\phi}}} \right)}} & (16) \\{H_{\phi} = {\frac{{- j}\; Y_{0\;{eff}}}{k_{eff}\mu}\left( {{\mu\frac{\delta\; E_{z}}{\delta\;\rho}} + {\frac{{- j}\;\kappa}{\rho}\frac{\delta\; E_{z}}{\delta\;\phi}}} \right)}} & (17) \\{k_{eff} = {\omega\sqrt{ɛ\;\mu_{eff}}}} & (18) \\{Y_{0\;{eff}} = \sqrt{ɛ\;\text{/}\mu_{eff}}} & (19) \\{\mu_{eff} = {\left( {\mu^{2} - \kappa^{2}} \right)\text{/}\mu}} & (20)\end{matrix}$

Solving Equation (13), another relation between the axial electric fieldand the transverse magnetic field can be obtained as given by Equation(21).

$\begin{matrix}{{\frac{1}{\rho}\left\lbrack {\frac{\delta\left( {\rho\; H_{\phi}} \right)}{\delta\rho} - \frac{\delta\; H_{\rho}}{\delta\phi}} \right\rbrack} = {j\;{\omega ɛ}\; E_{z}}} & (21) \\{{{\rho^{2}\frac{\delta^{2}E_{z}}{{\delta\rho}^{2}}} + {\rho\frac{\delta\; E_{z}}{\delta\rho}} + {\rho^{2}k_{eff}^{2}E_{z}} + \frac{\delta^{2}E_{z}}{{\delta\phi}^{2}}} = 0} & (22)\end{matrix}$

Substituting by Equations (16) and (17) in Equation (21) yields Equation(22). This equation is the same differential equation obtained whilesolving the TM mode in the cylindrical waveguide and the solution takesthe following form in Equation (23).

$\begin{matrix}{E_{zn} = {A_{n}{J_{n}\left( {k_{eff}\rho} \right)}e^{{jn}\;\phi}}} & (23) \\{H_{\phi n} = {{- {jY}_{eff}}A_{n}{R_{n}^{h}\left( {k_{eff}\rho} \right)}e^{{jn}\;\phi}}} & (24) \\{{R_{n}^{h}\left( {k_{eff}\rho} \right)} = {{J_{n}^{\prime}\left( {k_{eff}\rho} \right)} + {\frac{{ng}_{\kappa}}{k_{eff}\rho}{J_{n}\left( {k_{eff}\rho} \right)}}}} & (25) \\{{R_{n}^{e}\left( {k_{eff}\rho} \right)} = {J_{n}\left( {k_{eff}\rho} \right)}} & (26)\end{matrix}$

Only Bessel functions of the first kind are considered as the solutionhas to be finite at P=0, whilst Bessel functions of the second kind goto 1 at this point. Substituting into Equation (17), an expression of H_can be obtained in Equation (24) where R_(n) ^(h)(k_(eff)ρ) is given byEquation (25). Keeping in mind that the mode order n can take positiveand negative values, the previous function defines the magnetic fielddistribution in p direction for two counter-rotation of odes, while thecorresponding function for the electric field is a Bessel function ofthe first kind and it can be written as Equation (26).

The previous expressions describe all possible modes inside the ferriteresonator. The field inside this resonator is the summation of allpossible modes. The fields can be expressed as Equations (27) and (28).

$\begin{matrix}{E_{z} = {\sum\left\lbrack {A_{n}{R_{n}^{e}\left( {k_{eff}\rho} \right)}e^{{jn}\;\phi}} \right\rbrack}} & (27) \\{H_{\phi} = {{- {jY}_{eff}}{\sum\left\lbrack {A_{n}{R_{n}^{h}\left( {k_{eff}\rho} \right)}e^{{jn}\;\phi}} \right\rbrack}}} & (28) \\{{\sin(\psi)} = \frac{W}{2a}} & (29)\end{matrix}$

The circulators schematic configuration is shown in FIG. 2 whilst thesimulated model of an ideal RGW circulator is depicted in FIGS. 3A to 3Hrespectively. FIGS. 3A and 3B illustrate the simulated model for acirculator according to an embodiment of the invention exploitingstraight arms in top view and three-dimensional (3D) model,respectively, whilst FIGS. 3C to 3F illustrate the simulated modelaccording to an embodiment of the invention with curved arms. FIGS. 3Gand 3H depict the simulated model for a circulator according to anembodiment of the invention exploiting two curved arms inthree-dimensional (3D) model respectively focusing on the PerfectElectrical Conductor (PEC) and Perfect Magnetic Conductor (PMC) boundaryconditions respectively.

The coupling area at each port is determined by the port width W and thecoupling angle. These can be related to each other by Equation (29). Thebasic function of the circulator is to couple all the input power fromport 1 to port 2. The required field distribution for this functionalityhas to have a Poynting vector in −â_(p) at Ø=0 and a Poynting vector inâ at Ø=Ø_(p2)=120° (at the surface of the resonator p=a). The isolationcondition of the circulator means that no power is coupled to port 3.This can be achieved only if both fields vanish at Ø=Ø_(p3)=240°. Therequired field distribution to satisfy the circulator conditions can bestated mathematically by Equations (30) and (31) with the conditionsdefined by Equations (32) and (33).

$\begin{matrix}{{E_{z,{req}}\left( {a,\phi} \right)} = \left\{ \begin{matrix}E_{z\; 1}^{out} & {{- \psi} < \phi < \psi} \\E_{z\; 2}^{out} & {{{2\pi\text{/}3} - \psi} < \phi < {{2\pi\text{/}3} + \psi}} \\0 & {{{4\pi\text{/}3} - \psi} < \phi < {{4\pi\text{/}3} + \psi}}\end{matrix} \right.} & (30) \\{{H_{\phi,{req}}\left( {a,\phi} \right)} = \left\{ \begin{matrix}H_{z\; 1}^{out} & {{- \psi} < \phi < \psi} \\H_{z\; 2}^{out} & {{{2\pi\text{/}3} - \psi} < \phi < {{2\pi\text{/}3} + \psi}} \\0 & {{{4\pi\text{/}3} - \psi} < \phi < {{4\pi\text{/}3} + \psi}}\end{matrix} \right.} & (31) \\{E_{z\; 1}^{out} = {- E_{z\; 2}^{out}}} & (32) \\{H_{\phi 1}^{out} = H_{\phi 2}^{out}} & (33)\end{matrix}$

If the field distribution has satisfied the previous conditions, thePoynting vectors at port 1 and port 2 are given by Equations (33A) and(33B) respectively. The natural behavior of the resonator leads tohaving a zero tangential magnetic field at the disc surface. The discsurface can be approximately modeled as PMC surface due to therelatively high dielectric constant of ferrites (ε_(r)>10 for most offerrites). This point will be revisited later as this condition is verycritical in the circulator design procedure. In order to apply thecirculator boundary condition of the magnetic field, the magnetic fieldoutside the disc should be obtained. The field distribution outside theferrite disc is basically a function of the feeding structure of thecenter junction. In the case of the stripline both fields can beexpressed by Equations (34) and (35) respectively.

$\begin{matrix}{\mspace{79mu}{{\overset{\_}{P}}_{1} = {{E_{z\; 1}^{out}{\hat{a}}_{z} \times H_{Ø1}^{out}\mspace{11mu}{\hat{a}}_{Ø}} = {{- {\hat{a}}_{p}}P_{0}}}}} & \left( {33A} \right) \\{\mspace{79mu}{{\overset{\_}{P}}_{2} = {{{- E_{z\; 1}^{out}}{\hat{a}}_{z} \times H_{Ø1}^{out}\mspace{11mu}{\hat{a}}_{Ø}} = {{\hat{a}}_{p}P_{0}}}}} & \left( {33B} \right) \\{E_{z} = \left\{ {\begin{matrix}{- {\sum\limits_{m\mspace{14mu}{odd}}{\frac{2}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{14mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\phi} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}}} & (A) \\{\sum\limits_{m\mspace{14mu}{odd}}{\frac{2}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{14mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\phi} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{p}}\left( {h - z} \right)} \right)}} & (B) \\{\sum\limits_{m\mspace{14mu}{odd}}{\frac{2}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{14mu}\cosh\mspace{11mu}\left( {m\;\pi\;{h/W_{p}}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\left( {\phi - \frac{2\pi}{3}} \right)} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}} & (C) \\{- {\sum\limits_{m\mspace{14mu}{odd}}{\frac{2}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{14mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\left( {\phi - \frac{2\pi}{3}} \right)} \right)\mspace{11mu}\cosh\;\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}}} & (D)\end{matrix}\begin{matrix}{{{(A)\mspace{14mu} 0} > z > {- h}},{\psi > \phi > {- \psi}}} & {{{(C)\mspace{14mu} 0} > z > {- h}},{{\frac{2\pi}{3} + \psi} > \phi > {\frac{2\pi}{3} - \psi}}} \\{{{(B)\mspace{14mu} h} > z > 0},{\psi > \phi > {- \psi}}} & {{{(D)\mspace{14mu} h} > z > 0},{{\frac{2\pi}{3} + \psi} > \phi > {\frac{2\pi}{3} - \psi}}}\end{matrix}} \right.} & (34) \\{E_{z} = \left\{ {\begin{matrix}{- {\sum\limits_{m\mspace{14mu}{odd}}{\frac{2Y_{0}}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{14mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\phi} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}}} & (A) \\{\sum\limits_{m\mspace{14mu}{odd}}{\frac{2Y_{0}}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{14mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\phi} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{p}}\left( {h - z} \right)} \right)}} & (B) \\{- {\sum\limits_{m\mspace{14mu}{odd}}{\frac{2Y_{0}}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)\mspace{11mu}\cos\mspace{11mu}\left( {\phi - \frac{2\pi}{3}} \right)}{ɛ\mspace{14mu}\cosh\mspace{11mu}\left( {m\;\pi\;{h/W_{p}}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\left( {\phi - \frac{2\pi}{3}} \right)} \right)\mspace{11mu}{\cosh\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}}}} & (C) \\{\sum\limits_{m\mspace{14mu}{odd}}{\frac{2Y_{0}}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)\cos\mspace{11mu}\left( {\phi - \frac{2\pi}{3}} \right)}{ɛ\mspace{14mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}{\cos\left( {\frac{m\;\pi}{2\sqrt{3}}\left( {\phi - \frac{2\pi}{3}} \right)} \right)}\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}} & (D)\end{matrix}\begin{matrix}{{{(A)\mspace{14mu} 0} > z > {- h}},{\psi > \phi > {- \psi}}} & {{{(C)\mspace{14mu} 0} > z > {- h}},{{\frac{2\pi}{3} + \psi} > \phi > {\frac{2\pi}{3} - \psi}}} \\{{{(B)\mspace{20mu} h} > z > 0},{\psi > \phi > {- \psi}}} & {{{(D)\mspace{14mu} h} > z > 0},{{\frac{2\pi}{3} + \psi} > \phi > {\frac{2\pi}{3} - \psi}}}\end{matrix}} \right.} & (35)\end{matrix}$

The detailed derivation of Equations (34) and (35) is given in AppendixA. This implements the existence of the magnetic field at ports 1 and 2while ensuring the nonexistence of the magnetic field at port 3. Thisideal field boundary condition can be satisfied only by having aninfinite number of modes. Comparing Equations (28) and (35) theamplitude A_(n) can be expressed by Equation (36) where x=k_(eff)a. Bystraight forward manipulation, exactly at the middle of the structure,it can be obtained given by Equation (36) and (37) where I₁(ψ, m, n) isdefined by Equation (38).

$\begin{matrix}{\mspace{79mu}{A_{n} = {j{\frac{1}{Y_{0{eff}}}\left\lbrack \frac{1}{{J_{n}^{\prime}(x)} + {\frac{{ng}_{\kappa}}{x}{J_{n}(x)}}} \right\rbrack}*\frac{1}{2\pi}{\int_{- \pi}^{\pi}{H_{\phi}^{out}e^{{- {jn}}\;\phi}d\;\phi}}}}} & (36) \\{A_{n} = {j{\frac{1}{Y_{0{eff}}}\left\lbrack \frac{1 + e^{{- j}\; 2\pi\;{n/3}}}{{J_{n}^{\prime}(x)} + {\frac{{ng}_{\kappa}}{x}{J_{n}(x)}}} \right\rbrack}*\left( \frac{2Y_{0}}{ɛ} \right){\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m}*{I_{1}\left( {\psi,m,n} \right)}}}}} & (37) \\{\mspace{79mu}{{I_{1}\left( {\psi,m,n} \right)} = {\int_{- \psi}^{\psi}{\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\phi} \right)\mspace{11mu}{\cos(\phi)}e^{{- j}\mspace{11mu} n\;\phi}d\;\phi}}}\ } & (38)\end{matrix}$

The effect, characteristic admittance can be written as Equations (39)and (40) where η₀=√{square root over (μ₀/∈₀)} and the effectivecharacteristic impedance can be expressed as Equation (41).

$\begin{matrix}{{1\text{/}Y_{0{eff}}} = {\sqrt{\mu_{eff}\text{/}ɛ} = {\eta_{{eff}_{r}} \cdot \eta_{0}}}} & (39) \\{x = {k_{eff}a}} & (40) \\{\eta_{{eff}_{r}} = \sqrt{\frac{\mu_{eff}\text{/}\mu_{0}}{ɛ_{r}}}} & (41)\end{matrix}$

Substituting into Equation (27), the electric field can be written asgiven by Equation (42). The electric field at the port can be obtainedby calculating the average value of this field in the interval aroundthe port while substituting at p=a. This can be expressed as Equation(43) where ϕ_(P)=120° or 240°. Performing the integral the averageelectric field at the port yields Equation (44). Equation (44) can thenbe written in a compact form as given by Equation (45) where f_(n)(x,g_(k)) is given by Equation (46).

$\begin{matrix}{E_{z} = {\sum{j\;{\eta_{{eff}_{r}} \cdot {\eta_{0}\left\lbrack \frac{\left( {1 + e^{{- j}\; 2\pi\;{n/3}}} \right){J_{n}\left( {k_{eff}(\rho)} \right)}}{{J_{n}^{\prime}(x)} + {\frac{{ng}_{\kappa}}{x}{J_{n}(x)}}} \right\rbrack}} e^{j\; n\;\phi}*\left( \frac{2Y_{0}}{ɛ} \right){\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m}*{I_{1}\left( {\psi,m,n} \right)}}}}}} & (42) \\{\mspace{79mu}{E_{zp} = {\frac{1}{2\psi}{\int_{\phi_{p} - \psi}^{\phi_{p} + \psi}{E_{z}d\;\phi}}}}}_{\rho = a} & (43) \\{E_{z} = {\sum{j\;{\eta_{{eff}_{r}} \cdot {{\eta_{0}\left( \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{n\;\psi} \right)}\left\lbrack \frac{\left( {1 + e^{{- j}\; 2\pi\;{n/3}}} \right){J_{n}\left( {k_{eff}(\rho)} \right)}}{{J_{n}^{\prime}(x)} + {\frac{{ng}_{\kappa}}{x}{J_{n}(x)}}} \right\rbrack}} e^{j\; n\;\phi}*\left( \frac{2Y_{0}}{ɛ} \right){\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m}*{I_{1}\left( {\psi,m,n} \right)}}}}}} & (44) \\{E_{z} = {\sum{j\;{\eta_{{eff}_{r}} \cdot {{\eta_{0}\left( \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{n\;\psi} \right)}\left\lbrack \frac{\left( {1 + e^{{- j}\; 2\pi\;{n/3}}} \right)}{f_{n}\left( {x,g_{\kappa}} \right)} \right\rbrack}} e^{j\; n\;\phi}*\left( \frac{2Y_{0}}{ɛ} \right){\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m}*{I_{1}\left( {\psi,m,n} \right)}}}}}} & (45) \\{\mspace{79mu}{{f_{n}\left( {x,g_{\kappa}} \right)} = {\frac{J_{n}^{\prime}(x)}{J_{n}(x)} + \frac{{ng}_{\kappa}}{x}}}} & (46)\end{matrix}$

The previous equations are obtained by applying the boundary conditionsof the tangential magnetic fields only. In most of the published work,this is a turning point, where the junction impedance is evaluated. Theequations till this point contain three unknowns g_(k), x and ψ. Thevalue of g_(k) is related to the magnetic biasing point of the ferritedisc and the saturation magnetic value of the used ferrite. The othertwo parameters (x and ψ) determine the physical dimensions of thecirculator. These two parameters refer to the ferrite disc radius andthe stripline width respectively. It is normal to assume the value ofg_(k) as a first step of the design. This value should be selected from0 to −1 for below resonance mode of operation. Actually, it should be inbetween −0.2 and −0.8 to avoid losses due to resonance mode or below thesaturation mode. This choice will be addressed in details subsequently.The design procedure established by the inventors is based on the belowresonance mode of operation. However, it can be modified to design inthe above resonance mode. The majority of the circulator designerswithin the prior continue with their design procedures based uponcalculating the junction impedance and equating the imaginary part ofthe junction impedance by zero to ensure the isolation in the design. Infact, this provides one equation in two unknowns, x and ψ: Thiscomplicates the procedure to reach the required dimension of thecirculator.

Accordingly, to the inventors, it is evident that there are missingboundary conditions to be applied. The electric field at the input portshould have the same magnitude of the coupled port with a 180° phaseshift, while the field at the isolated port should be equal to zero.These conditions can be listed as Equations (47) and (48) respectively.After straightforward manipulations, two governing equations can beobtained to satisfy the circulator conditions. These two equations willbe referred to as the “Circulator Ergodic Equations” and are given byEquations (49) and (50), respectively.

$\begin{matrix}{{{\mspace{79mu} E_{{zp}\; 1}}_{\phi = 0} = {- E_{{zp}\; 2}}}}_{\phi = \frac{2\pi}{3}} & (47) \\{{\mspace{79mu} E_{{zp}\; 3}}_{\phi = {\frac{4\pi}{3} = \frac{{- 2}\pi}{3}}} = 0} & (48) \\{{\sum\limits_{n = {- \infty}}^{\infty}{\left\lbrack \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{n\;\psi} \right\rbrack\frac{\cos\left( {2\pi\; n\text{/}3} \right)}{f_{n}\left( {x,g_{\kappa}} \right)}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m}*{I_{1}\left( {\psi,m,n} \right)}}}}} = 0} & (49) \\{{\sum\limits_{n = {- \infty}}^{\infty}{\left\lbrack \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{n\;\psi} \right\rbrack\frac{1}{f_{n}\left( {x,g_{\kappa}} \right)}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,n} \right)}}}}} = 0} & (50)\end{matrix}$

The integral can be evaluated as it is illustrated in Appendix A. Inorder to simplify the solution of the ergodic equations, uniformexcitation at the coupling area can be utilized to yield the simplerexpressions in Equations (51) and (52).

$\begin{matrix}{{\sum\limits_{n = {- \infty}}^{\infty}{\left\lbrack \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{n\;\psi} \right\rbrack\frac{\cos\;\left( {2\pi\; n\text{/}3} \right)}{f_{n}\left( {x,g_{\kappa}} \right)}}} = 0} & (51) \\{{\sum\limits_{n = {- \infty}}^{\infty}{\left\lbrack \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{n\;\psi} \right\rbrack\frac{1}{f_{n}\left( {x,g_{\kappa}} \right)}}} = 0} & (52)\end{matrix}$

Using the ergodic equations, the physical dimensions of the circulatorcan be obtained through solving two nonlinear equations in two unknowns.In order to achieve this, one of several iterative techniques can beused, where the initial values of both variables can be selected withrelative ease. The expected value of ψ should be less than π/3, hence agood starting point is π/6 in the iterative solution. In some cases, theiterative algorithm fails to find the required solution because of theinitial point. In such cases, the initial guess should be changed andthe iterative algorithm has to be repeated.

3. Evaluation of Design Procedures within the Prior Art

The previous analysis yielded the ergodic equations of the junctioncirculators. These equations can be used to evaluate prior art designprocedures. The analysis presented in this section can be divided intotwo categories. The first category is based on considering only threeterms of the series n=0, 1 and −1, while the second category ofcirculator design procedures had taken into consideration up to n=±3. Inthe following part, both techniques are going to be criticized based onthe previous analysis. Hence, an overview of the recent progress in thecirculator analysis and design is discussed.

3.1 Low Magnetic Bias Junction Circulators

This design procedure was originally by Bosma in “On StriplineY-Circulation at UHF” (IEEE Microwave Theory and Techniques, Vol. 12,No. 1, pp. 61-72) and was further developed by Fay and Comstock in “Onthe Theory of the Ferrite Junction Circulator” (Int. Symp. ProfessionalTechnical Group on Microwave Theory and Techniques, Vol. 64, No. 1, pp.54-59) and “Operation of the Ferrite Junction Circulator” (IEEEMicrowave Theory and Techniques, Vol. 13, No. 1, pp. 15-27). Theseresearchers assumed that all terms with n>1 have a small contributionand can be neglected. In this design procedure, three terms only of theseries are considered, n={0, 1, −1}. The selected solution is the onethat gives resonance under no bias. This resulted in {grave over(J)}₁(k_(eff)a)=0 as mentioned in their analysis. To evaluate thevalidity of this assumption, the inventors consider only the first threeterms in Equations (49) and (50). These can be written as Equations (53)and (54) respectively.

$\begin{matrix}{0 = {\frac{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,0} \right)}}}{\frac{J_{0}^{\prime}(x)}{J_{0}(x)}} - \frac{\left( \frac{1}{2} \right)\left( \frac{\sin(\psi)}{\psi} \right)^{2}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,1} \right)}}}}{\frac{J_{1}^{\prime}(x)}{J_{1}(x)} + \frac{g_{\kappa}}{x}} - \frac{\left( \frac{1}{2} \right)\left( \frac{\sin(\psi)}{\psi} \right)^{2}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,{- 1}} \right)}}}}{\frac{J_{- 1}^{\prime}(x)}{J_{- 1}(x)} + \frac{g_{\kappa}}{x}}}} & (53) \\{0 = {\frac{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,0} \right)}}}{\frac{J_{0}^{\prime}(x)}{J_{0}(x)}} - \frac{\left( \frac{\sin(\psi)}{\psi} \right)^{2}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,1} \right)}}}}{\frac{J_{1}^{\prime}(x)}{J_{1}(x)} + \frac{g_{\kappa}}{x}} - \frac{\left( \frac{\sin(\psi)}{\psi} \right)^{2}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,{- 1}} \right)}}}}{\frac{J_{- 1}^{\prime}(x)}{J_{- 1}(x)} + \frac{g_{\kappa}}{x}}}} & (54)\end{matrix}$

It can be proved that I₁(ψ, m, 1)=I₁(ψ, m, −1). Hence, using theidentities in Table 2, the previous expressions can be written asEquations (55)/(56) and (57)/(58) respectively.

$\begin{matrix}{\mspace{79mu}{0 = {\left( \frac{J_{0}(x)}{J_{1}(x)} \right){\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,0} \right)}}}}}} & (55) \\{{+ \left( \frac{\sin(\psi)}{\psi} \right)^{2}}\frac{{J_{1}^{\prime}(x)}\text{/}{J_{1}(x)}}{\left( \frac{J_{1}^{\prime}(x)}{J_{1}(x)} \right)^{2} - \left( \frac{g_{\kappa}}{x} \right)^{2}}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,1} \right)}}}} & (56) \\{\mspace{79mu}{0 = {\left( \frac{J_{0}(x)}{J_{1}(x)} \right){\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,0} \right)}}}}}} & (57) \\{{- 2}*\left( \frac{\sin(\psi)}{\psi} \right)^{2}\frac{{J_{1}^{\prime}(x)}\text{/}{J_{1}(x)}}{\left( \frac{J_{1}^{\prime}(x)}{J_{1}(x)} \right)^{2} - \left( \frac{g_{\kappa}}{x} \right)^{2}}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,1} \right)}}}} & (58)\end{matrix}$

TABLE 2 Some Bessel Function Identities Expression Equivalent J₀′(x) −J₁(x) J_(−n)(x) (−1)^(n) J_(−n)(x) J_(−n)′(x) (−1)^(n) J_(−n)′(x)J_(−n)′(x)/J_(−n)(x) J_(−n)′(x)/J_(−n)(x)

Hence, both expressions can be written as Equations (59) and (60).

$\begin{matrix}{\left\lbrack {J_{1}^{\prime}(x)} \right\rbrack^{2} + {\left( \frac{\sin(\psi)}{\psi} \right)^{2}\left( \frac{J_{1}^{2}(x)}{J_{0}(x)} \right){\quad{{\frac{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,1} \right)}}}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,0} \right)}}}{J_{1}^{\prime}(x)}} - {\quad{{\left( \frac{g_{\kappa}}{x} \right)^{2}\left( {J_{1}(x)} \right)^{2}} = 0}}}}}} & (59) \\{\left\lbrack {J_{1}^{\prime}(x)} \right\rbrack^{2} - {2\mspace{11mu}\left( \frac{\sin(\psi)}{\psi} \right)^{2}\left( \frac{J_{1}^{2}(x)}{J_{0}(x)} \right)\frac{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,1} \right)}}}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,0} \right)}}}{J_{1}^{\prime}(x)}} - {\quad{{\left( \frac{g_{\kappa}}{x} \right)^{2}\left( {J_{1}(x)} \right)^{2}} = 0}}} & (60)\end{matrix}$

It is very clear that these two equations are contradicting. They cannotbe solved together. However, the solution provided by Bosma was {graveover (J)}₁(x)=0 which leads to have x=1.841. It is recommended in thedesign based on their solution is to select a small value forg_(κ)(=k/μ). This can be predicted easily by a quick look to theprovided solution. As long as {grave over (J)}₁(x)=0 the condition thatshould be satisfied is given by Equation (61). By substituting x=1.841we obtain Equation (62).

$\begin{matrix}{{\left( \frac{g_{\kappa}}{x} \right)^{2}\left( {J_{1}(x)} \right)^{2}} = 0} & (61) \\{{0.099\mspace{11mu} g_{\kappa}^{2}} = 0} & (62)\end{matrix}$

This condition will never be satisfied, which means that consideringthree terms only are not a suitable solution for the problem. On theother hand, by having a small value for k/μ, the design results in anoperating circulator even with a poor performance. That is why they haverecommended a range of k/μ to be less than 0.5. The insufficient numberof terms in the analysis resulted in inaccurate approximation. This factwas discovered within the prior art, but without a proper explanation.In the following subsection, the modified design methodologies byconsidering seven terms of the series is criticized.

3.2 Tracking and Semi-Tracking Solution for Junction Circulators

The presented work by Bosma, Fay, and Comstock was modified subsequentlyby Helszajn in “Synthesis of Quarter-Wave Coupled Circulators withChebyshev Characteristics” (IEEE Microwave Theory and Techniques, Vol.20, No. 11, pp. 764-769) to include up to n=±3. This design procedurewas developed within the prior art for several cases by Helszajn. In theanalysis provided by Helszajn, the magnetic field boundary conditionswere applied before the port impedance is calculated and equated to zeroat the center frequency of the design. After assuming g_(k), this leadsto have one equation in two unknowns, x and ψ. Helszajn also splits thecirculator design into two groups: the weekly magnetized (small valuefor k/μ) and the tracking circulator (higher values for k/μ). Althoughthe work performed by Helszajn provides a design with improvedperformance in the then prior art it still suffers limitations such asevaluating only seven terms which deteriorates the accuracy of theprovided solution. Moreover, the solution does not pay attention to therealistic field distribution at the feeding structure. Further, thereare two other limitations as the methodology has a single equation intwo unknowns and is based on the assumption of having a zero tangentialmagnetic field outside the disc. The realistic magnetic field, however,is attenuated radially outside the disc, but is non-zero. This can beimplemented by a modified Bessel function in the regions among theports. The fringing fields result in a larger electrical radius of thedisc, which shifts the measured frequency response with respect to thedesign. In many stripline circulators, the stripline has to be filledwith a dielectric. Whilst this aids in achieving the required impedancematching and provides a mechanical support for the stripline it violatesmore and more of the PMC assumptions at the boundary between the ferritedisc and the surrounding material.

3.2.1 Modified Seven Term Based Solution

A simple modification can be performed by just considering the firstseven terms in the series in the ergodic equation. Accordingly, applyingin these equations n=0, ±1, ±2, ±3 the expressions given in Equations(63) and (64) are obtained where the function T_(n)(ψ, x, g_(k)) isdefined as a twin function to consider two opposite rotating modes ofthe same order as defined in FIG. 65). Solving Equations (63) and (64)together leads to Equations (66) and (67).

$\begin{matrix}{{{T_{0}\left( {\psi,x,g_{\kappa}} \right)} - {\frac{1}{2}{T_{1}\left( {\psi,x,g_{\kappa}} \right)}} - {\frac{1}{2}{T_{2}\left( {\psi,x,g_{\kappa}} \right)}} + {T_{3}\left( {\psi,x,g_{\kappa}} \right)}} = 0} & (63) \\{{{T_{0}\left( {\psi,x,g_{\kappa}} \right)} + {T_{1}\left( {\psi,x,g_{\kappa}} \right)} + {T_{2}\left( {\psi,x,g_{\kappa}} \right)} + {T_{3}\left( {\psi,x,g_{\kappa}} \right)}} = 0} & (64) \\{{T_{n}\left( {\psi,x,g_{\kappa}} \right)} = \left\{ \begin{matrix}\begin{matrix}{\left( \frac{\sin\left( {n\;\psi} \right)}{\psi} \right)\left( \frac{2{J_{n}^{\prime}(x)}\text{/}{J_{n}(x)}}{\left( {{J_{n}^{\prime}(x)}\text{/}{J_{n}(x)}} \right)^{2} - \left( {{ng}_{\kappa}\text{/}x} \right)^{2}} \right)} \\\frac{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,1} \right)}}}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}*{I_{1}\left( {\psi,m,0} \right)}}}\end{matrix} & {n \neq 0} \\{{- {J_{0}(x)}}\text{/}{J_{1}(x)}} & {n = 0}\end{matrix} \right.} & (65) \\{\mspace{79mu}{{{T_{1}\left( {\psi,x,g_{\kappa}} \right)} + {T_{2}\left( {\psi,x,g_{\kappa}} \right)}} = 0}} & (66) \\{\mspace{79mu}{{{T_{0}\left( {\psi,x,g_{\kappa}} \right)} + {T_{3}\left( {\psi,x,g_{\kappa}} \right)}} = 0}} & (67)\end{matrix}$

In spite of that the previous equations consider the only 7 terms of theseries (n=0, ±1, ±2, ±3) but, it is able to provide a better accuracy.Here, the stripline field distribution in the coupling area areconsidered. Moreover, the solution to the provided equations isstraightforward as it is based on solving two nonlinear equations in twounknowns.

3.3 Recent Developments in Circulators

During the last decade whilst there have been many publications withinthe prior art these have predominantly addressed deploying circulatorsin different applications or introduced new performance evaluationmethodologies. Within this, many researchers are directed to implementthe circulator functions through active elements. Yet, in contrast,relatively minimal attention has been given to the actual development ofthe basic analysis of the circulators despite the massive increase intheir deployment with the penetration of wireless technologies intonearly every aspect of our lives. The junction circulators utilized inthe recent publications are predominantly based upon the traditionalmethods presented 50 years ago. Within the preceding sections whilst theinventors have given an indication of the major research directions inthis field in recent years and whilst there is work directed to theanalysis of the circulator, see for example Porranzl et al. in “A NewActive Quasi-Circulator Structure with High Isolation for 77-GHzAutomotive FMCW Radar Systems in SiGe Technology” (IEEE CompoundSemiconductor Integrated Circuit Symposium 2015, pp. 1-4) and Kim et al.in “Three Octave Ultra-Wideband 3-Port Circulator in 0.11 mm CMOS”(Electronics Letters, Vol. 49, No. 10, pp. 648-650). However, even thisprior art is more focused on enhancing the old analysis by includingsome new parameters rather than addressing and revising the major stepsin the analysis.

4. The Junction Immittance & Scattering Matrix

The junction circulator is a N-port network, however, the most commonlyused configuration is the three port circulator, N=3. The port electricfield and magnetic field based on the previous analysis supra can beexpressed as Equations (68) and (69) respectively. The previous equationof the magnetic field at the port is obtained through the integration asdefined in Equation (70) and the electric and magnetic fields arerelated to each other through the matrix relationship given in Equation(71). Based on symmetry the statement in Equation (72) can be written.

$\begin{matrix}{E_{zp} = {\sum\limits_{n}{{{jn}_{{eff}_{r}} \cdot {{n_{0}\left( \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{\psi} \right)}\left\lbrack \frac{1 + e^{{- j}\frac{2\pi\; n}{3}}}{f_{n}\left( {x,g_{\kappa}} \right)} \right\rbrack}}{e^{{jn}\;\phi_{p}} \cdot {\quad{\left( \frac{2Y_{o}}{ɛ} \right){\sum\limits_{m\mspace{14mu}{odd}}{\frac{\left( {\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)} \right.}{2m\;\pi^{2}}{I_{1}\left( {\psi,m,n} \right)}}}}}}}}} & (68) \\{\mspace{79mu}{H_{\phi\; p\; 1} = {\left( \frac{2Y_{o}}{ɛ} \right){\sum\limits_{m\mspace{14mu}{odd}}^{\;}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{2m\;\pi^{2}}{I_{1}\left( {\psi,m,0} \right)}}}}}} & (69) \\{\mspace{79mu}{H_{\phi\; p} = {\frac{1}{2\psi}{\int_{\phi_{p} - \psi}^{\phi_{p} - \psi}{H_{\phi}d\;\phi}}}}}_{\rho = \alpha} & (70) \\{\mspace{79mu}{\begin{bmatrix}E_{{zp}\; 1} \\E_{{zp}\; 2} \\E_{{zp}\; 3}\end{bmatrix} = {\begin{bmatrix}\eta_{11} & \eta_{12} & \eta_{13} \\\eta_{21} & \eta_{22} & \eta_{23} \\\eta_{31} & \eta_{32} & \eta_{33}\end{bmatrix}\begin{bmatrix}H_{\phi\; p\; 1} \\H_{\phi\; p\; 2} \\H_{\phi\; p\; 3}\end{bmatrix}}}} & (71) \\{\mspace{79mu}{\eta_{11} = {\eta_{22} = \eta_{23}}}} & (72)\end{matrix}$

In addition, for the circulator operation, the cyclic symmetryrequirement is a necessary condition. This can be formalized asEquations (73) and (74). This reduces the number of unknowns in thepreviously mentioned matrix representations to the given Equations (75)and (76).

$\begin{matrix}{\eta_{12} = {\eta_{31} = \eta_{23}}} & (73) \\{\eta_{13} = {\eta_{21} = \eta_{32}}} & (74) \\{\left\lbrack E_{zp} \right\rbrack = {\left\lbrack \eta_{3 \times 3} \right\rbrack\left\lbrack H_{\phi\; p} \right\rbrack}} & (75) \\{\begin{bmatrix}E_{{zp}\; 1} \\E_{{zp}\; 2} \\E_{{zp}\; 3}\end{bmatrix} = {\begin{bmatrix}\eta_{11} & \eta_{12} & \eta_{13} \\\eta_{13} & \eta_{11} & \eta_{12} \\\eta_{12} & \eta_{13} & \eta_{11}\end{bmatrix}\begin{bmatrix}H_{\phi\; p\; 1} \\H_{\phi\; p\; 2} \\H_{\phi\; p\; 3}\end{bmatrix}}} & (76)\end{matrix}$

The intrinsic impedance matrix elements can be calculated from Equations(77) to (79) respectively. For the circulator to operate in the requiredmanner, the electric field and the magnetic field at the isolated port,both, have to be equal to zero simultaneously. The input intrinsicimpedance of the junction can be written as Equation (80).

$\begin{matrix}{\eta_{11} = {\sum{\left\lbrack {j\frac{{\eta_{eff} \cdot \eta_{0}}\psi}{\pi}} \right\rbrack\left( \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{\psi} \right)\frac{1}{f_{n}\left( {x,g_{k}} \right)}\frac{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{m\;\pi}*{I_{1}\left( {\psi,m,n} \right)}}}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{m\;\pi}*{I_{1}\left( {\psi,m,0} \right)}}}}}} & (77) \\{\eta_{12} = {\sum{\left\lbrack {j\frac{{\eta_{eff} \cdot \eta_{0}}\psi}{\pi}} \right\rbrack\left( \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{\psi} \right)\frac{e^{{- j}\; 2n\;{\pi/3}}}{f_{n}\left( {x,g_{k}} \right)}\frac{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{m\;\pi}*{I_{1}\left( {\psi,m,n} \right)}}}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{m\;\pi}*{I_{1}\left( {\psi,m,0} \right)}}}}}} & (78) \\{\eta_{13} = {\sum{\left\lbrack {j\frac{{\eta_{eff} \cdot \eta_{0}}\psi}{\pi}} \right\rbrack\left( \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{\psi} \right)\frac{e^{{- j}\; 4n\;{\pi/3}}}{f_{n}\left( {x,g_{k}} \right)}\frac{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{m\;\pi}*{I_{1}\left( {\psi,m,n} \right)}}}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( {m\;\pi\text{/}2\sqrt{3}} \right)\psi} \right)}{m\;\pi}*{I_{1}\left( {\psi,m,0} \right)}}}}}} & (79) \\{\mspace{79mu}{\eta_{in} = {\eta_{11} - {\eta_{12}^{2}\text{/}\eta_{13}}}}} & (80)\end{matrix}$

To determine the impedance representation of the junction, the term η₀has to be replaced by z₀, where z₀ represents the characteristicimpedance of the input transmission line at the junction port. This alsoprovides the normalized impedance representation relative to the portimpedance.

As the junction design aims to satisfy the ergodic equations at theoperating center frequency, the frequency response of the junctionadmittance should be considered to obtain the matching within theobjective bandwidth. Solving the ergodic circulator equations results inhaving g_(k), ψ and x. This defines the material specifications and thedimensions. Hence, the input impedance of this specified ferrite disccan be plotted with respect to frequency in order to design the suitablematching network. The scattering parameters can be obtained through theimpedance representation through Equation (81).

$\begin{matrix}{\lbrack S\rbrack = {\left( {{\frac{Z_{r}}{\eta_{0}}\lbrack\eta\rbrack} + \lbrack I\rbrack} \right)^{- 1}\left( {{\frac{Z_{r}}{\eta_{0}}\lbrack\eta\rbrack} - \lbrack I\rbrack} \right)}} & (81)\end{matrix}$

5. Design Procedure for Stripline Circulators

The circulator specifications are defined by the operating frequencyband and the required matching and isolation levels. Starting with thesespecifications, the design frequency is calculated. The inventor'sdesign procedure splits the process into two major steps. The first stepis to solve at the design frequency, which results in selecting therequired parameters for which the ergodic circulator equations aresatisfied. Then, the second step is to design the matching transformerto cover the required bandwidth. The deployed matching methodology ofthe inventors is to change the characteristic impedance of the feedingstructure through a dielectric filling. This matching technique is ableto provide both the required matching but also contributes to coolingthe ferrite discs which in turns allow for increased the overall powerhandling capability of the structure. Finally, the necessary effectivepermittivity for the structure is achieved through the use of aperforated substrate. The perforations allowed a standard substratedielectric constant to be reduced to the desired design value. Further,embodiments of the invention place the design frequency offset from themiddle frequency in between the two band edges as the fringing fieldswithin the structure result in an increased effective “electrical”diameter of the center disc relative to its physical diameter. Withinthe prior art, such fringing fields have typically been ignored. Incontrast, the design methodology of the inventors employs three designconsiderations to compensate for their presence and effects.

First, the inventors employ a surrounding dielectric material that has adielectric constant less than 60% of the ferrite dielectric constant.This decreases to some extent the fringing fields outside the ferritedisc. As the inventor's selected matching technique depends uponchanging the relative permittivity of the filler material for thestripline this requirement places, typically, an upper maximum value forthe relative permittivity.

Second, the design frequency is typically set to a frequency higher thanthe middle frequency of the target operating frequency range via ascaling factor, e.g. 5% such thatf _(CENTRE)=ScalingFactor×((f _(LOWER) +f _(UPPER))/2)=1.05×((f _(LOWER)+f _(UPPER))/2).

Third, the hole/recess within the structure is larger than the ferritedisc in order to introduce an air gap around the ferrite disc between itand the surrounding material. This reduces the fringing fieldssignificantly within the portion of the ferrite disc within thesurrounding medium. Within the prior art, the design goal has been theelimination of any air gaps to ensure good contact between the ferriteand the surrounding medium. The American National Standards Institute(ANSI) within Standard ANSI 4.1 defines the tolerance for aninterference fit (Class V) between a hole and shaft for a nominal 0.125″(3.175 mm) diameter has the hole specified with 0.1244″±0.0004″(3.160±0.010 mm) and the shaft specified as 0.1252″±0.0002″ (3.180±0.005mm). However, even these standard design tolerances are below that oftypical Computer Numerical Control (CNC) cutting tools within commercialmachine shops which are typically are typically within the range of0.001″ (approx. 25 μm) to 0.0005″ (approx. 13 μm). Rather the inventorsre-formulate this to being that the goal is to ensure full contactbetween the ferrite disc and the electrical transmission lines at theports of the circulator.

Taking these considerations into account, the circulators designed bythe inventors meet the required circulator specifications. The analysisof the ferrite resonator can be modified to include the fringing fieldsoutside the ferrite resonator (disc) by representing these fringingfields with a modified Bessel function and applying the appropriateboundary conditions. The mathematical formulation can be modified, but,practically, it is extremely hard to ensure the contact between the discand the surrounding material over the whole perimeter as noted supra fortypical high volume commercial machining tolerances. Accordingly, theinventors reformulated design forces the existence of air and includedit in the design analysis.

5.1 Step 1: The Ferrite Disc Design

The first step aims to identify the center material fully. The outcomeof this step is to choose the required magnetic saturation point of theferrite material, i.e.: the value of 4πM_(s) as well as the appliedexternal magnetic field. Then, the disc radius and the coupling angleare determined. This is performed through the following procedure.

Step 1A: The design frequency can be obtained from both band edges,following the considerations described supra, through Equation (82)wherein the scaling factor offsets the design center frequency. Forexample, the inventors within the design, analysis employScalingFactor=1.05, i.e. a 5% offset.f _(CENTRE)=ScalingFactor×((f _(LOWER) +f _(UPPER))/2)  (82)

Step 1B: Next the ferrite material is selected. This begins with theassumption of the Gytropy g_(k), which the inventors have established as0.2≤g_(K)≤0.8, and a magnetic biasing ratio α. Based upon these themagnetic saturation is calculated from the Gytropy value using Equation(7). The assumed value of α defines the magnetic biasing circuit afterbuilding the circulator, as it relates the magnetic saturation of theferrite with the applied the magnetic field H₀(O_(e))=*α. The inventorswithin their analysis typically limit this to 0≤α≤0.5 as higher valuesresult in above saturation losses whilst negative values result in lowfield losses. Generally, in the magnetic biasing circuit design, thedemagnetization factor should be taken into account. This concept isaddressed within the prior art. In the examples presented the inventorshave employed a thin disc physical configuration, which has anapproximate unity demagnetization factor. Accordingly, no detailed studywill be given to this concept here, however, it is worth to mention thatconsidering this factor accurately, should result in a better design.The inventors being the design process from a certain α as a startingpoint, typically around 0.2. As discussed supra, assuming this parameteris equal to zero as within the prior art, is not a practical assumptionas the DC magnetic field responsible of biasing the ferrite is generatedby a permanent magnet. The magnetic field associated with this magnetcannot be controlled in a continuous manner, especially, when there areother constraints such as the behavior with temperature or aging effectsof the used magnets. In this specification the inventors perform thedesign analysis twice, for α=0 and α=0.2

Step 1C: After calculating 4πM_(s) and H₀, the full permeability tensorcan be obtained based on the expressions in Table 1. Accordingly, it isrelatively straightforward to obtain μ_(eff) and k_(eff).

Step 1D: Accordingly, the ergodic circulator equations can be solvedyielding values of ψ and x can be obtained. Since x=k_(eff), the radiusof the ferrite disc is calculated.

Step 1E: Based upon the radius and the coupling angle the striplinewidth, can be calculated through Equation (29). Within this analysis,the height of the ferrite disc equals to the height of the stripline.This height can be selected keeping in mind that h_(f)<a. This conditionis required in order to ensure small axial variations. It should benoticed that the analysis of the ferrite disc started with neglectingthe axial variation. A valid assumption employed by the inventors intheir design is that h_(f)=0.4*a.

Accordingly, by working through the previously described points, theferrite disc for the circulator is fully determined. The material isselected based on the required magnetic saturation value; then theradius is obtained. Finally, the height is assumed within a reasonablerange.

5.2 Step 2: Matching Section Design

The second step within the inventor's method according to an embodimentof the invention is to design the feeding structure of the junction.This is considered as a matching network design problem and the selectedmatching methodology employed here, is simply to perform the matching atthe center frequency by the dielectric filling of the stripline.

Step 2A: The directly attached stripline to the ferrite disc has a widthW that is calculated in the previous steps. Through the width and theheight, the air filled stripline characteristics impedance can becalculated as given by Equation (83) after Pozar where W₀/b is the ratiobetween the effective width of the line to the total stripline heightand b≈2h_(f). The total stripline height is equal to the summation ofthe top and the bottom ferrite discs neglecting the stripline thickness.This effective ratio is calculated from Equation (84).

$\begin{matrix}{Z_{0} = \frac{30\pi}{{W_{e}\text{/}b} + 0.441}} & (83) \\{\frac{W_{e}}{b} = \left\{ \begin{matrix}\frac{W}{b} & {{{for}\mspace{14mu} W\text{/}b} > 0.35} \\{\frac{W}{b} - \left( {0.35 - \frac{W}{b}} \right)^{2}} & {{{for}\mspace{14mu} W\text{/}b} < 0.35}\end{matrix} \right.} & (84)\end{matrix}$

Step 2B: The input resistance of the junction is calculated at thedesign frequency. As discussed supra, the imaginary part has to vanishas long as the ergodic circulator equations are satisfied. This reducesthe input admittance at the center frequency to be that given inEquation (85). At the center frequency, the circulator conditions arefully satisfied. This results in the condition given by Equation (86)being obtained. This reduces the input resistance equation to be thatgiven by Equation (87).

$\begin{matrix}{\mspace{79mu}{R_{in} = {{Re}\mspace{11mu}\left\{ {\frac{Z_{0}}{\eta_{0}}\left( {\eta_{11} - {\eta_{12}^{2}\text{/}\eta_{13}}} \right)} \right\}}}} & (85) \\{\mspace{79mu}{\eta_{11} = {- \eta_{13}}}} & (86) \\{R_{in} = {{Re}\mspace{11mu}\left\{ {\sum{\left\lbrack {j\frac{{\eta_{eff} \cdot \eta_{0}}\psi}{\pi}} \right\rbrack\left( \frac{\sin\mspace{11mu}\left( {n\;\psi} \right)}{\psi} \right)\frac{1 + e^{j\; 2\; n\;{\pi/3}}}{f_{n}\left( {x,g_{k}} \right)}\left( \frac{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{m\;\pi}*{I_{1}\left( {\psi,m,n} \right)}}}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{m\;\pi}*{I_{1}\left( {\psi,m,0} \right)}}} \right)}} \right.}} & (87)\end{matrix}$

Step 2C: The matching is achieved through a dielectric filling for thestripline. It is important to note that in most cases there is anessential constraint related to the available dielectric constants. Theclosest available material should be selected for the most feasiblematching. It is worth mentioning that the selected standard value forthe relative permittivity has to be higher than the design value.

Step 2D: The realization of the final design is carried out throughperforming perforation inside the selected standard substrate. Throughthis process, the effective relative permittivity can be extremely closeto the design value.

Step 2E: The input admittance frequency dependency has to be considered.This can be found via plotting the input admittance of the junctionversus frequency for the selected material within the operatingbandwidth. This gives an indication of the expected performance of thematching network.

Step 2F: For the final assessment of the design the scatteringparameters can be calculated, over the whole frequency band, fromEquation (88).

$\begin{matrix}{\lbrack S\rbrack = {\left( {{\frac{Z_{r}}{\eta_{0}}\lbrack\eta\rbrack} + \lbrack I\rbrack} \right)^{- 1}\left( {{\frac{Z_{r}}{\eta_{0}}\lbrack\eta\rbrack} - \lbrack I\rbrack} \right)}} & (88)\end{matrix}$It is important to note that using this methodology, the port impedanceis forced by the selection of the junction impedance and the couplingangle. This may result in nonstandard values of the striplinecharacteristic impedance. If the design has to be connected through astandard line, another matching transformer has to be introduced inbetween the current value and the required standard. This can alsoachieve by further perforation.

6. Circulator Design Examples

Within this section the inventor's present two designs, one for the Kband and the other in X-band.

6.1 Example 1: K-Band Circulator with Perforated Matching Network 6.1ADesign #1: α=0

In the first example, the frequency band extends from 18 GHz to 26.5GHz, where the center frequency is at 22.25 GHz. The ferrite material isassumed to have a relative permittivity of “f=12:5. The nominal valuesfor these materials are in the range of 12 to 16.

Step 1: The design frequency in this case is f_(d)=23.363 GHz

Step 2: Assume g_(k)=−0.6, and starting with the exact saturation withα=0. The selected material in this case has a magnetic saturation of4πM_(s)=5006G. This can be evaluated through Equation (89).

$\begin{matrix}{{4\pi\; M_{s}} = {\frac{f_{d}}{2.8*10^{6}}p_{m}}} & (89)\end{matrix}$

Step 3: The magnetic bias in this case is H_(ext)=4πM_(s) this resultsin H₀=0. The values of μ_(eff) and k_(eff) are 0.64μ₀ H/m and 1384rad/m, respectively.

Step 4: The solution of the ergodic equations provides the values of xand ψ to be 1.5727 and 0.5344, respectively.

Step 5: The ferrite disc radius can be calculated from x=k_(eff) a. Thedisc radius in this case is a=0.0447 inches (1.135 mm).

Step 6: The stripline width is W=0.0456 inch (1.158 mm);

Step 7: The selected height of the ferrite disc is h=0.4 a=0.0179 inches(0.455 mm).

Step 8: Based on the selected dimensions, the air filled stripline has acharacteristic impedance of Z_(0 air filled)=146.57Ω

Step 9: The input resistance at resonance is R_(in)=43.571Ω

Step 10: The required dielectric constant for the matching is largerthan 11 which is not accepted based on the design considerationmentioned previously. The highest recommended value isε_(d,max)=0.6*ε_(f)=7.5. This maximum value will be used in the designof this circulator.

Step 11: The scattering parameters are calculated and compared with thesimulated results. The simulated model is shown in FIGS. 3A to 3H withboth configurations. FIGS. 3A and 3B depict the simulated model for acirculator according to an embodiment of the invention exploitingstraight arms in top view and three-dimensional (3D) model,respectively, whilst FIGS. 3C to 3F depict the simulated model accordingto an embodiment of the invention with curved arms. FIGS. 3G and 3Hdepict the simulated model for a circulator according to an embodimentof the invention exploiting two curved arms in three-dimensional (3D)model respectively focusing on the Perfect Electrical Conductor (PEC)and Perfect Magnetic Conductor (PMC) boundary conditions respectively.As evident from the Figures, the circulator can be configured with threestraight arms or with one straight and two curved arms or othercombinations. The waveguide arm curvature is performed to have all portsaligned with the reference planes. This configuration of the circulatoris practically preferred in some systems to be connected with differentcomponents. FIG. 4 shows a good agreement between the analytical and thesimulated response. In this case, the simulated model considers thedielectric constant of the filling material to be exactly equal to thedesign value regardless the realization of this value.

Step 12: Finally, Rogers TMM 10 standard substrate is selected with arelative permittivity of 9.2. The perforation is performed on thissubstrate to obtain an effective dielectric constant of 7.5, which isthe design value. The perforation is defined by two parameters, the holediameter, and the hole separation. The designed values for bothparameters are 0.018 inches and 0.038 inches, respectively. The designprocedure of the perforation will be discussed in a separate section.The final configuration is shown in FIGS. 5A and 5B, respectively, withthe corresponding response illustrated in FIG. 6.

6.1B Design #1: α=0.2

The same example can be solved again, changing the assumption of α. Inthis solution, the value of α is assumed to be 0.2.

In this case, the value of the saturation magnetization of the materialis changed to be 4πM_(s)=4635.4 G, while the internal magnetic field isH₀=927.1O_(e). The effective permeability and the effective wave numberare changed and this will affect the radius to be a=0.0475 inches (1.207mm). The width of the stripline is W=0.0484 inches (1.229 mm). Theferrite disc height is H_(f)=0.019 inches (0.483 mm). The selecteddielectric constant is still ε_(d)=7.5, due to the same reasonsdiscussed before. The results of this example are shown in FIG. 7, wherethere is an excellent agreement between the analytical model and thesimulated model with the effective relative permittivity for thematching section. FIG. 8 illustrates the response final circulatorresponse fabricated with a standard Rogers TMM10 with perforation.

6.2 Example 2: X-Band Circulator with Perforated Matching Network

In the second example, the frequency band extends from 8.2 GHz to 12.4GHz. The used ferrite material is assumed to have the same relativepermittivity of ε_(f)=12.5.

Both designs for this example follow the previously described procedure.This procedure results can be summarized in Table 3. The comparisonsbetween the analytical model response and the simulated response areshown in FIGS. 9 and 10 for both designs. In the previous two figures,the relative permittivity of the filling material is selected with anonstandard value, ∈_(r)=7.5. The final step is to realize thenonstandard value of the dielectric constant through performingperforation inside a standard Rogers TMM 10 substrate. The sameperforation parameters of Example 1 for the K-band circulator aredeployed. The number of holes increases as the overall size of thecirculator in this example is almost doubled. FIG. 11 shows the top viewand the 3D view of the simulated model of the X-band circulator withperforation. The final simulated results of the X-band circulatorresponse with perforation are shown in FIGS. 12 and 13.

TABLE 3 X-Band Circulator Designs based on Inventive Design ProcedureParameter Design 1 ^(α = 0) Design 2 ^(α = 0.2) Design Frequency ^(f)^(d) 10.815 GHz 10.815 GHz Ferrite Saturation 2317.5 G  2145.8Magnetization ^(4πM) ^(s) Magnetic Biasing Ratio ^(α) 0 0.2 EffectivePermeability ^(μ) ^(eff) 0.640^(μ) ^(o) ^(H/m) 0.568^(μ) ^(o) ^(H/m)Effective Wave Number ^(k) ^(eff) 640.66 603.55 rad/m Gytropy ^(g) ^(K)0.6 0.6 Ergodic Equation Solution for ^(x) 1.5727 1.5727 ErgodicEquation Solution for ^(ψ) 0.5344 0.5344 Ferrite Disc Radius ^(a)0.0966″ (2.454 mm) 0.1026″ (2.606 mm) Stripline Width ^(W) 0.0980″(2.489 mm) 0.1045″ (2.654 mm) Ferrite Disc Height ^(h) ^(f) 0.0387″(0.983 mm) 0.0410″ (1.041 mm) Stripline Filling 7.5 7.5 RelativePermittivity ^(ε) ^(d)

6.3. Perforation Design

An important step within the innovative design methodology is thereplacement of the ideally simulated matching section with aconventional perforated substrate that has a very close relativepermittivity value to the assumed one. The implementation of a certainvalue of the relative permittivity using a standard substrate of ahigher relative permittivity is discussed in many applications like thedielectric resonator antennas and reflectarrays, see for exampleHelszajn et al. in ““Fringing Effects in Re-Entrant and InvertedRe-Entrant Turnstile Waveguide Junctions using Cylindrical Resonators”(IET Microwaves, Antennas & Propagation, Vol. 5, No. 9, pp. 1109-1115).Many equations relate between the perforation dimensions and theeffective relative permittivity. Equation (90) is described in a lot ofwork related to reflectarrays, e.g. Moeine-Fard et al. in “InhomogeneousPerforated Reflect-Array Antennas” (Wireless Engineering and Technology,2011) to provide this relation where d_(h) and g_(h) are the holediameter and the gap between two adjacent holes respectively. Theminimum possible hole diameter is a limiting factor determined by theavailable machining facility, while the maximum hole diameter consideredhas to ensure that the medium will act in a homogenous way. The maximumhole diameter value should be less than one tenth of the wavelength atthe max frequency within the operating bandwidth. The required initialvalue of the hole diameter is obtained from the previous equation. AsEquation (90) is related to the reflect array problems, where the waveis normally incident on the top face of the perforated substrate, theselected values of the hole diameters should be verified and tuned toachieve the required dielectric constant. A numerical extraction for therelative permittivity is performed by simulating a parallel platewaveguide filled with the perforated substrate, then the relativepermittivity is extracted from the phase of S₂₁ as the mode propagatingin this case is pure TEM mode. Through this simple simulation, theperforation parameters can be tuned to achieve the design value of theeffective dielectric constant.

$\begin{matrix}{ɛ_{eff} = {{ɛ_{r}\left( {1 - {\frac{\pi}{2\sqrt{3}}\left( \frac{d_{h}}{d_{h} + g_{h}} \right)^{2}}} \right)} + {\frac{\pi}{2\sqrt{3}}\left( \frac{d_{h}}{d_{h} + g_{h}} \right)^{2}}}} & (90)\end{matrix}$

6.4 The Field Distribution

The objective for the selected design parameters is to satisfy thecirculator conditions. These conditions can be depicted from FIGS. 14Aand 14B, where the axial field distribution is plotted in one case ofeach example. As it can be observed clearly that the field isconcentrated at port 1 and port 2 while no field is penetrating port 3.This last port is considered as the isolation port. It is evident inthis figure also that there is some small field distribution outside theferrite disc. This leads to some discrepancy between the analyticalmodel and the simulated results.

6.5 Examples 3 & 4: 5G 15 GHz and 30 GHz Circulators with Ridge GapWaveguide Matching Networks

In this section a detailed design procedure is presented for thecirculator designs which follows essentially the same procedure as thatoutlined in Section 6.1 for the K-band circulator (and employed inSection 6.2 for the X-band circulator) is presented based upon a RidgeGap Waveguide (RGW) circulator with ideal Boundary conditions such asdepicted in FIGS. 3E to 3H respectively. In this instance, the RGW isimplemented with ideal PMC boundaries.

Step 1: Assuming the Gytropy value in the range between −1 to 0 forbelow resonance mode of operation. It is recommended to be limited inbetween −0.2 and −0.8 to avoid resonance losses and below saturationlosses. These facts are addressed within the prior art. Whilst it ispossible also to select positive values of the Gytropy for the highresonance mode of operation the prior art denotes that such devicesusually have smaller operating bandwidths. The saturation magnetizationof the ferrite disc can be calculated through Equation (89).

Step 2: The input reactance is equated to zero to find the ferriteradius and the coupling angle.

Step 3: The ferrite height is assumed, where h_(f)≤a (assumingh_(f)=0.4a). This condition ensures the neglected axial variations asassumed in the analysis. This is a design parameter, which means thatthis height can take many other values as long as the axial variation isneglected.

Step 4: The input resistance is calculated at the design frequency.

Step 5: Design the matching network to connect the junction to thefeeding line. It should be taken into consideration that thepermittivity of the RGW should be smaller than the ferrite permittivity.As such, a PMC boundary assumption at the perimeter of the ferrite discbetween the ports is a valid approximation. A roughly estimated limit isthat ε_(d,max)=0.8ε_(f), where the dielectric filling of the RGW has arelative permittivity of ε_(d), while the ferrite relative permittivityis ε_(f). There are no definite limits for the maximum value of thedielectric permittivity deployed for matching. However, the PMC boundaryassumption is increasingly violated when both values are getting closer.After calculating the required value of the relative permittivity forthe matching, a lower value should be considered. The objective is theRGW circulator design, which has more fringing fields around the ridgecompared to the model with PMC surface. This results in having a largereffective electrical width than the physical width. The characteristicimpedance of RGW will be smaller than the ideal model. To compensate forthis phenomenon, in advance, the utilized dielectric constant is lowerthan the required one by 30%.

Step 6: Finally, the analytically predicted response of the scatteringparameters has to be compared to the simulated response based on idealboundary conditions.

Step 7: RGW circulator realization, wherein the objective of this stepis to replace the PMC boundary around the ridge with the periodic cells.To achieve that, the following two-step procedure is performed:

-   -   Step 7A: Design the periodic cell with the same gap height with        a bandgap that contains the circulator operating bandwidth.    -   Step 7B: Replace the ideal PMC surface with the designed cells.        Hence, further optimization should be performed to obtain the        required response, if needed.

As mentioned supra the design frequencies are f_(d1)=15 GHz andf_(d1)=30 GHz. The assumed Gytropy is chosen to be −0.6. This value is adesign parameter, i.e. it can be selected with any value in thespecified range. The saturation magnetization of the ferrite iscalculated using Equation (89) to be 4πM_(S)=3214.3G; 6428.6G,respectively. The dielectric constant of the ferrite materials should beobtained from the ferrite provider where the nominal value of thisparameter is typically between 12 and 15. An example of a ferrite beingTT1-3000 which has a specified ε_(f)=12.9 The ferrite materials can becustom made with a specific 4πM_(S) or can be ordered with standardvalues. If the required material is not available, the designer can usea material with slightly lower 4πM_(S) and obtain the same Gytropy byextra magnetization. It is assumed here that the required value isavailable.

The imaginary part of the input impedance is equated to zero. Thisresults in a coupling angle ψ=0.5344 radians for both designs andferrite disc radii are a₁=0.0686″ (1.742 mm) and a₁=0.0343″ (0.871 mm).Hence, the ferrite heights can be assumed to beh_(f1)=0.4*0.0686=0.0274″ (0.696 mm) and h_(f1)=0.4*0.0343=0.0137″(0.348 mm). It is important to notice that, both, the coupling angle andthe disc radius, determine the ridge width. This width W_(r) can becalculated through Equation (29) 6, where W_(r1)=0.0699″ (1.775 mm) andW_(r1)=0.0349″ (0.886 mm). The input resistance of the ferrite disc isequal to 16.82Ω in both cases, but the characteristic impedance of theair filled RGW is 54.98Ω. The RGW characteristic impedance is calculatedapproximately through the stripline Equation (83).

The required dielectric constant to ensure the desired matching is above10. As mentioned before, the selected filling material of ε_(d)=7.75,which is around 30% below the required value for the matching. As notedpreviously the simulated model of the RGW circulator is depicted inFIGS. 3E to 3H where FIG. 3G focuses on the top PEC boundary condition,and FIG. 3H shows the PMC boundary. The PMC boundary is considered onlyto obtain the ideal RGW response. The comparison between the analyticaland the simulated response are depicted in FIGS. 15A and 15B for the 15GHz and 30 GHz designs. The circulator response is considered in a 40%bandwidth centered at the design frequency. It is evident from FIGS. 15Aand 15B that the designs are shifted down in frequency, intentionally.This shift is due to the deployed matching section. The dielectricconstant used in matching is less than the required value in order tocompensate for the expected change in the characteristic impedance ofthe ridge. Both designs cover more than 25% bandwidth with a matchingand isolation levels of −15 dB, while the proposed bandwidths for 5Gmobile applications do not exceed 10%. In addition, the realization ofthis design by RGW will compensate for the frequency shift, which willincrease the circulator bandwidth. This is explained in the followingdescription.

The periodic cells deployed in these RGW designs are the traditional“bed of nails” unit cell (BNUC). These types of cells can achieve abandwidth better than 2:1 and in some applications, when the operatingbandwidth exceeds 2.5:1, other types of unit cells have to be used. Thedesign of the BNUC is well covered within the prior art and is depictedin FIGS. 16A and 16B, whilst the Eigenmode solutions are depicted inFIGS. 16C and 16D, The cell dimensions are the cell width W_(C), the pinwidth W_(P), the gap height h_(g) and the pin height h_(p). The valuesof these parameters are listed in Table 4 for both designs, where thegap height of the cell is precisely equal to the ferrite disc height. Itis evident from FIGS. 16C and 16D that the stop band of the designedcells covers wider bandwidths in both cases that required.

The cells are placed surrounding the ridge to keep the PMC boundaryconditions assumed in the ideal deign. The circulator configuration isshown in FIG. 17, where the upper ground is removed to show thestructure details. The responses of the final designs are presented inFIGS. 18A and 18B. It is evident from these figures that the curves areshifted up again. The practical ridge implementations have more fieldfringing around the ridge compared to the ideal PMC boundaries, whichincreases the effective width of the ridge. The wider ridge reduces thecharacteristic impedance of the feeding line, which compensates thecharacteristic impedance of the matching network. The final designcovers, almost, a 40% bandwidth with a matching level of −15 dB. Itwould be evident that other different matching techniques or multiplestage matching networks can be employed to provide wider operatingbandwidths without departing from the design and device constructionmethodologies according to embodiments of the invention.

TABLE 4 Bed of Nail Unit Cell Dimensions Dimension Design 1 Value(inch/mm) Design 2 Value (inch/mm) Cell width, ^(W) ^(C) 0.1813″/4.605mm 0.0515″/2.499 mm Pin width, ^(W) ^(P) 0.0976″/2.479 mm 0.0343″/0.871mm Gap height, ^(h) ^(g) 0.0279″/0.709 mm 0.0137″/0.348 mm Pin height^(h) ^(p) 0.1813″/4.605 mm 0.0984″/2.499 mm

7. Circulator Power Handling

The power handling of any microwave device is measured by two majorparameters, the maximum power, and the average power. Regarding themaximum power that can be handled by a circulator according to theinventive design methodology, then it is limited only by the striplinefeeding structure at the height of the structure is kept the same. Thematching element deployed in the matching section design is thedielectric filling, and the gap height is not utilized as a tuningelement. Moreover, the average power that can be handled by thisstructure is increased as a direct effect of the dielectric filling ofthe stripline. The dielectric filling helps in transferring the heatgenerated inside the ferrite disc away by conduction. Most of the heatgenerated inside the ferrite disc is located at the high field spots.These areas are in a direct contact with the dielectric filling of thefeeding lines.

8. Options

Within the embodiments of the invention described and depicted supra thedielectric material disposed on the electrically non-conductive andferromagnetic material, e.g. ferrite, is assumed to be air. However, itwould be evident that in other embodiments of the invention thedielectric material may be another material with low dielectric constantsuch as an inert gas, e.g. nitrogen, argon, carbon dioxide, etc., or aninert filler material such as a xerogel/aerogel or spin-on polymer suchas Teflon/PTFE provided that the required design requirements can bemet.

Similarly, the dielectric within the outer portion of the microwavecirculator providing the microwave matching circuit may be composed of astandard solid dielectric, a ridge gap waveguide (or multi-ridgewaveguide) comprising metallic insertions on the top and/or bottomwalls, an RGW waveguide, which is implemented with a “bed of nails” oralternatively it may employ a microwave waveguide employing a pluralityof posts with a first predetermined material and diameter embedded witha filler of a second predetermined material.

APPENDIX A

1: The Stripline Electric and Magnetic Field Distribution

FIG. 14 illustrates the stripline configuration at port 1, which isaddressed in the following analysis. Starting from Laplace equation,Equation (91) in the transverse direction where the boundary conditionsare defined by Equation (92).∇_(t)Φ(y,z)=0  (91)Φ(y,z)=0 y=±W _(t)/2  (92)Φ(y,z)=0 z=±h _(f)  (93)

Although the real model does not have PEC boundaries in the y−direction, but the total width W_(t) is very large compared to thestripline width W. As W_(t)>>W, the effect of the hypothetical PEC sideboundaries will be neglected. Solving Laplace equation and applying theboundary conditions, the expression in Equation (94) is obtained. Thenthe axial electric field can be obtained through

$E_{z} = \frac{\partial}{\partial z}$(Φ) to be that given in Equation (95).

$\begin{matrix}{{\Phi\left( {y,z} \right)} = \left\{ \begin{matrix}{\sum\limits_{m\mspace{14mu}{odd}}{B_{m}\mspace{11mu}\cos\mspace{11mu}\left( {\frac{m\;\pi}{W_{t}}y} \right)\mspace{11mu}\sinh\mspace{11mu}\left( {\frac{m\;\pi}{W_{t}}\left( {z - h_{f}} \right)} \right)}} & {0 > z > {- h_{f}}} \\{\sum\limits_{m\mspace{14mu}{odd}}{B_{m}\mspace{11mu}\cos\mspace{11mu}\left( {\frac{m\;\pi}{W_{t}}y} \right)\mspace{11mu}\sinh\mspace{11mu}\left( {\frac{m\;\pi}{W_{t}}\left( {h_{f} - z} \right)} \right)}} & {h_{f} > z > 0}\end{matrix} \right.} & (94) \\{E_{z} = \left\{ \begin{matrix}{\sum\limits_{m\mspace{14mu}{odd}}{B_{m}\mspace{11mu}\cos\mspace{11mu}\left( {\frac{m\;\pi}{W_{t}}y} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{t}}\left( {z - h_{f}} \right)} \right)}} & {0 > z > {- h_{f}}} \\{\sum\limits_{m\mspace{14mu}{odd}}{B_{m}\mspace{11mu}\cos\mspace{11mu}\left( {\frac{m\;\pi}{W_{t}}y} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{t}}\left( {h_{f} - z} \right)} \right)}} & {h_{f} > z > 0}\end{matrix} \right.} & (95) \\{\mspace{79mu}{B_{m} = \frac{2W_{t}\mspace{11mu}\sin\mspace{11mu}\left( {m\;\pi\; W\text{/}2W_{t}} \right)}{\left( {m\;\pi} \right)^{2}ɛ\mspace{11mu}\cosh\mspace{11mu}\left( {m\;\pi\; h_{f}\text{/}W_{t}} \right)}}} & (96)\end{matrix}$

Assuming uniform charge distribution along y-direction at the centerstrip ((W/2)>y>(−W/2),z=0) the constant B_(m) can be obtained as givenby Equation (96). As the propagating modes along the stripline are TEM,the magnetic field H_y can be obtained by multiplying the axial electricfield by Y_0 (the characteristic admittance). Hence, the tangentialfield on the disc surface H_Ø can be obtained from the expression of H_yby direct resolution. The axial electric field and the tangentialmagnetic field can be obtained as given by Equations (97) and (98)respectively.

$\begin{matrix}{E_{z} = \left\{ \begin{matrix}{- {\sum\limits_{m\mspace{14mu}{odd}}{\frac{2}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{11mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\phi} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}}} & (A) \\{\sum\limits_{m\mspace{14mu}{odd}}{\frac{2}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{11mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\phi} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{p}}\left( {h - z} \right)} \right)}} & (B) \\{\mspace{11mu}{\sum\limits_{m\mspace{14mu}{odd}}{\frac{2}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( {m\;{\pi/2}\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{11mu}\cosh\mspace{11mu}\left( {m\;\pi\; h\text{/}W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\left( {\phi - \frac{2\pi}{3}} \right)} \right)\mspace{11mu}{\cosh\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}}}} & (C) \\{- {\sum\limits_{m\mspace{14mu}{odd}}{\frac{2}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{11mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\left( {\phi - \frac{2\pi}{3}} \right)} \right)\mspace{11mu}{\cosh\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}}}} & (D)\end{matrix} \right.} & (97) \\{E_{z} = \left\{ \begin{matrix}{- {\sum\limits_{m\mspace{14mu}{odd}}{\frac{2Y_{0}}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{11mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\phi} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}}} & (A) \\{\sum\limits_{m\mspace{14mu}{odd}}{\frac{2Y_{0}}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{11mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\phi} \right)\mspace{11mu}\cosh\mspace{11mu}\left( {\frac{m\;\pi}{W_{p}}\left( {h - z} \right)} \right)}} & (B) \\{\mspace{11mu}{- {\sum\limits_{m\mspace{14mu}{odd}}\;{\frac{2Y_{0}}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)}{ɛ\mspace{11mu}\cosh\mspace{11mu}\left( {m\;\pi\; h\text{/}W_{p}} \right)}\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\left( {\phi - \frac{2\pi}{3}} \right)} \right)\mspace{11mu}{\cosh\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}}}}} & (C) \\{\sum\limits_{m\mspace{14mu}{odd}}{\frac{2Y_{0}}{m\;\pi}\frac{\sin\mspace{11mu}\left( {\left( \frac{m\;\pi}{2\sqrt{3}} \right)\psi} \right)\cos\mspace{11mu}\left( {\phi - \frac{2\pi}{3}} \right)}{ɛ\mspace{11mu}\cosh\mspace{11mu}\left( \frac{m\;\pi\; h}{W_{p}} \right)}\cos\mspace{14mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\left( {\phi - \frac{2\pi}{3}} \right)} \right)\mspace{11mu}\cosh\mspace{14mu}\left( {\frac{m\;\pi}{W_{p}}\left( {z - h} \right)} \right)}} & (D)\end{matrix} \right.} & (98) \\\begin{matrix}{{{(A)\mspace{14mu} 0} > z > {- h}},{\psi > \phi > {- \psi}}} & {{{(C)\mspace{14mu} 0} > z > {- h}},{{\frac{2\pi}{3} + \psi} > \phi > {\frac{2\pi}{3} - \psi}}} \\{{{(B)\mspace{14mu} h} > z > 0},{{- \psi} > \phi > {- \psi}}} & {{{(D)\mspace{14mu} h} > z > 0},{{\frac{2\pi}{3} + \psi} > \phi > {\frac{2\pi}{3} - \psi}}}\end{matrix} & \;\end{matrix}$

2. The Stripline Junction Circulator Integral

As each port boundary, can be estimated to exist in the range

${\frac{2\pi}{2} + \phi_{P}} > \phi > {\frac{2\pi}{2} - {\phi_{P}.}}$The value of

${\phi_{P} = 0},\frac{2\pi}{3},\frac{4\pi}{3}$for ports 1, 2, and 3, respectively. Based on this fact the ratio ofW/W₁ can be obtained to be 1/√{square root over (3)}.

$\begin{matrix}{{I_{1}\left( {\psi,m,n} \right)} = {\int_{- \psi}^{\psi}{\cos\mspace{11mu}\left( {\frac{m\;\pi}{2\sqrt{3}}\ \phi} \right)\mspace{11mu}\cos\mspace{11mu}(\phi)e^{{- {jn}}\;\phi}d\;\phi}}} & (99)\end{matrix}$

The foregoing disclosure of the exemplary embodiments of the presentinvention has been presented for purposes of illustration anddescription. It is not intended to be exhaustive or to limit theinvention to the precise forms disclosed. Many variations andmodifications of the embodiments described herein will be apparent toone of ordinary skill in the art in light of the above disclosure. Thescope of the invention is to be defined only by the claims appendedhereto, and by their equivalents.

Further, in describing representative embodiments of the presentinvention, the specification may have presented the method and/orprocess of the present invention as a particular sequence of steps.However, to the extent that the method or process does not rely on theparticular order of steps set forth herein, the method or process shouldnot be limited to the particular sequence of steps described. As one ofordinary skill in the art would appreciate, other sequences of steps maybe possible. Therefore, the particular order of the steps set forth inthe specification should not be construed as limitations on the claims.In addition, the claims directed to the method and/or process of thepresent invention should not be limited to the performance of theirsteps in the order written, and one skilled in the art can readilyappreciate that the sequences may be varied and still remain within thespirit and scope of the present invention.

What is claimed is:
 1. A microwave circulator comprising: a pair ofelectrically non-conductive and ferromagnetic elements with specificmagnetic saturation value (M_(s)) each having a predetermined thicknessand a predetermined diameter; an electrical conductor plane comprising aplurality of microwave tracks and a central circular pad to which eachmicrowave track is coupled at a predetermined location, each microwavetrack comprising a first portion adjacent the central pad and a secondportion extending from the first portion to a distal point; a lowerelectrical ground plane; an upper electrical ground plane; a firstdielectric disposed between the electrical conductor plane and the lowerelectrical ground plane and having a thickness determined in dependenceupon the predetermined thickness of the electrically non-conductive andferromagnetic elements and an opening determined in dependence upon thepredetermined diameter of the electrically non-conductive andferromagnetic elements; a second dielectric disposed between theelectrical conductor plane and the upper electrical ground plane andhaving a thickness determined in dependence upon the predeterminedthickness of the electrically non-conductive and ferromagnetic elementsand an opening determined in dependence upon the predetermined diameterof the electrically non-conductive and ferromagnetic elements; whereinthe openings within the first dielectric and second dielectric have adiameter establishing a predetermined air gap between the externalperiphery of an electrically non-conductive and ferromagnetic elementand their respective dielectric when the electrically non-conductive andferromagnetic element is centrally disposed with the opening; the firstportion of each microwave track is air filled microwave track; and thesecond portion of each microwave track is a dielectric filled microwavetrack.
 2. The microwave circulator according to claim 1, wherein thesecond portion of each microwave track is a matching transformer betweenthe impedance of the central portion and an external microwave circuitto be coupled to the microwave circulator.
 3. The microwave circulatoraccording to claim 1, wherein the pair of electrically non-conductiveand ferromagnetic elements are formed from a ferrite.
 4. The microwavecirculator according to claim 1, wherein the microwave tracks arestriplines or ridge gap waveguides.
 5. The microwave circulatoraccording to claim 1, wherein the second dielectric is comprised of aplurality of posts of a first predetermined material, and predetermineddiameter disposed in a predetermined pattern within a filler of a secondpredetermined material.
 6. The microwave circulator according to claim1, wherein the pair of electrically non-conductive and ferromagneticelements are formed from a ferrite; the microwave tracks are ridge gapwaveguides; and the second dielectric is comprised of a plurality ofposts of a first predetermined material, and predetermined diameterdisposed in a predetermined pattern within a filler of a secondpredetermined material.
 7. The microwave circulator according to claim1, wherein the pair of electrically non-conductive and ferromagneticelements are formed from a ferrite; the microwave tracks are striplines;and the second dielectric is comprised of a plurality of posts of afirst predetermined material, and predetermined diameter disposed in apredetermined pattern within a filler of a second predeterminedmaterial.
 8. A microwave circulator comprising: a set of three parallelelectrical planes wherein the middle electrical plane comprises aplurality of microwave tracks and a central region coupled to theplurality of microwave tracks and each outer electrical plane is aground plane; wherein a central portion enclosed by the set of threeparallel electrical layers comprises an inner region with electricallynon-conductive and ferromagnetic elements of predetermined lateraldimensions disposed between each outer electrical plane and the middleelectrical plane and an outer region filled with a first dielectricmaterial of low dielectric constant such that those portions of eachmicrowave track in this outer region form microwave feeds coupled to thecentral region of the middle electrical plane at predeterminedlocations; an outer portion enclosed by the set of three parallelelectrical layers is filled with a second dielectric material such thatthose portions of each microwave track in this outer portion formmicrowave matching networks between the portion of each microwave trackin the outer region of the central portion and an external microwavecircuit to be coupled to the distal ends of each microwave track fromthe central portion; the plurality of microwave tracks are striplines;and the second dielectric material is comprised of a plurality of postsof a first predetermined material, and predetermined diameter disposedin a predetermined pattern within a filler of a second predeterminedmaterial.
 9. The microwave circulator according to claim 8, wherein theouter region of the central portion is determined by providing apredetermined gap around each electrically non-conductive andferromagnetic element between it and the dielectric material in theouter portion.
 10. A microwave circulator comprising: a set of threeparallel electrical planes wherein the middle electrical plane comprisesa plurality of microwave tracks and a central region coupled to theplurality of microwave tracks and each outer electrical plane is aground plane; wherein a central portion enclosed by the set of threeparallel electrical layers comprises an inner region with electricallynon-conductive and ferromagnetic elements of predetermined lateraldimensions disposed between each outer electrical plane and the middleelectrical plane and an outer region filled with a first dielectricmaterial of low dielectric constant such that those portions of eachmicrowave track in this outer region form microwave feeds coupled to thecentral region of the middle electrical plane at predeterminedlocations; an outer portion enclosed by the set of three parallelelectrical layers is filled with a second dielectric material such thatthose portions of each microwave track in this outer portion formmicrowave matching networks between the portion of each microwave trackin the outer region of the central portion and an external microwavecircuit to be coupled to the distal ends of each microwave track fromthe central portion; the plurality of microwave tracks are ridge gapwaveguides; and the second dielectric material is comprised of aplurality of posts of a first predetermined material, and predetermineddiameter disposed in a predetermined pattern within a filler of a secondpredetermined material.
 11. A method of designing a microwave circulatorcomprising: 1) establishing simulation data by solving a predeterminedset of closed form equations at a predetermined frequency relating tothe electrical and magnetic fields with respect to an electricallynon-conductive and ferromagnetic element comprising a firstpredetermined portion of the microwave circulator with low dielectricconstant material based microwave waveguides coupling to theelectrically non-conductive and ferromagnetic element; and 2) designinga matching transformer to cover a predetermined bandwidth of operationin dependence upon the simulation data established in step (1) usinghigh dielectric constant substrate based microwave waveguides forming amatching network between the low dielectric constant material basedmicrowave waveguides coupling to the electrically non-conductive andferromagnetic element and an external microwave circuit coupled to themicrowave circulator, the simulation data comprising a set of physicalproperties of the electrically non-conductive and ferromagnetic element,a set of physical properties of a plurality of microwave ports forming asecond predetermined portion of the microwave circulator and a set ofelectrical properties of the microwave ports.
 12. The method accordingto claim 11, wherein the low dielectric constant material basedmicrowave waveguides are striplines; and the low dielectric constantmaterial is comprised of a plurality of posts of a first predeterminedmaterial, and predetermined diameter disposed in a predetermined patternwithin a filler of a second predetermined material.
 13. The methodaccording to claim 11, wherein the microwave waveguides are ridge gapwaveguides; and the low dielectric constant material is comprised of aplurality of posts of a first predetermined material, and predetermineddiameter disposed in a predetermined pattern within a filler of a secondpredetermined material.
 14. A method of designing a microwave circulatorcomprising: 1) establishing simulation data by solving a predeterminedset of closed form equations at a predetermined frequency relating tothe electrical and magnetic fields with respect to an electricallynon-conductive and ferromagnetic element comprising a firstpredetermined portion of the microwave circulator; and 2) designing amatching transformer to cover a predetermined bandwidth of operation independence upon the simulation data established in step (1), thesimulation data comprising a set of physical properties of theelectrically non-conductive and ferromagnetic element, a set of physicalproperties of a plurality of microwave ports forming a secondpredetermined portion of the microwave circulator and a set ofelectrical properties of the microwave ports.
 15. The method accordingto claim 14, wherein at least one of: the electrically non-conductiveand ferromagnetic element is formed from a ferrite; the microwave portsare striplines or ridge gap waveguides; the plurality of microwave portsare formed from a dielectric material is comprised of a plurality ofposts of a first predetermined material and predetermined diameterdisposed in a predetermined pattern within a filler of a secondpredetermined material; and the plurality of microwave ports are formedfrom a dielectric material is comprised of a plurality of holes ofpredetermined diameter filled with a first predetermined material anddisposed in a predetermined pattern within a filler of a secondpredetermined material.